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An equation of change becomes useful when you can explain its quantities, solve it under stated conditions and check what the result means. Choose the route closest to your question.
Full chapter index · Work the cooling example · Connected article directory
The gold standard of Differential Equations is the ability to turn a justified rule of change into a checked explanation of a changing quantity. That includes choosing conditions, solving or approximating the equation, retaining its domain and testing what changes when an assumption changes.
This subject root belongs to How X Works and continues The Gold Standard of Education | Learning That Transfers into Life. The existing How Mathematics Works | Differential Equations explains the broader mechanism. The Core Aim of Mathematics Mastery | Differential Equations provides the established mastery route. Here, worked cases connect those guides with the subject’s standards of reasoning and verification.
The numerical cases are original fictional teaching examples. Parameters and conditions are supplied for the calculation; they are not measurements from a real population, object or installation. Read every result together with its equation, domain and assumptions.
Full chapter index
Chapters 1–4 · Define and check
Chapters 5–8 · Solve and interpret
Chapters 9–11 · Examine the assumptions
Chapters 12–16 · Follow motion and computation
Chapters 17–19 · Transfer and evaluate
CHAPTER 1 OF 19 · Define and check
1. Ask what is changing before reaching for a method
A tank receives water while some drains away. A warm object cools towards its surroundings. A structure moves after a force is removed. The question in each case is how a present condition and a rule of change determine what happens next.
A differential equation relates an unknown function to one or more of its derivatives. Its solution is a function satisfying that relationship on a stated domain. Initial or boundary conditions add information that can select particular solutions, when the problem supports that selection.
The gold standard is a connected explanation: what the quantities mean, why the rate law is appropriate, how the solution was obtained and what checks support its interpretation. A correct-looking formula is only one part of that account.
This root develops that standard through original worked examples. The existing How Mathematics Works and Mathematics Mastery guides provide the mechanism and learning routes; the directory connects them with modelling, numerical methods, dynamics, control and applications.
Continue into: How Mathematics Works | Differential Equations.
CHAPTER 2 OF 19 · Define and check
2. Keep quantities, units and conditions inside the equation
Let M(t) describe an amount of material, measured in grams, with time in minutes. Then dM/dt has units of grams per minute. An equation such as dM/dt = q − kM requires q in grams per minute and k in reciprocal minutes.
This unit check is part of the reasoning. It distinguishes an amount from a concentration, a rate constant from a flow rate and elapsed time from a count of discrete events. A number such as 0.2 is incomplete until its role and units are established.
A model also needs a domain. A volume may be nonnegative and limited by tank capacity. A time interval may end when an input changes. Solving the equation for every real t does not make every mathematical value meaningful for the physical question.
Write the assumptions alongside the model: constant inflow, well-mixed contents, constant surroundings or negligible friction, as appropriate. Those statements explain which changes would require a different equation rather than another round of algebra.
Continue into: How Units Work in Equations | Quantities, Conversion Factors, Derived Units and the Algebra of Measurement · How Dimensional Consistency Works | How to Check Equations Before You Trust the Calculation.
CHAPTER 3 OF 19 · Define and check
3. Distinguish order, linearity and the independent variables
An ordinary differential equation involves derivatives with respect to one independent variable. A partial differential equation involves partial derivatives of a function with several independent variables, such as temperature depending on position and time.
The order is the highest derivative appearing. A first-order equation may govern an amount; a second-order equation can describe displacement through velocity and acceleration. The meaning comes from the model, rather than from order alone.
A linear equation is linear in the unknown function and its derivatives, with coefficients allowed to depend on the independent variable. For example, y′ + t y = t² is linear, while y′ = y² is nonlinear. A coefficient depending on t does not by itself make the equation nonlinear.
These distinctions help choose methods. A technique valid for a first-order linear equation need not apply to a nonlinear equation. Classify the relationship before manipulating symbols, and retain any restrictions introduced during the manipulation.
Continue into: The Gold Standard of Calculus.
CHAPTER 4 OF 19 · Define and check
4. Verify a solution before trusting its appearance
Consider the original teaching problem y′ = 4t − 3 with y(0) = 7. Integrating gives y = 2t² − 3t + C. Applying the condition gives C = 7, so the proposed solution is y(t) = 2t² − 3t + 7.
There are two separate checks. Differentiation gives y′ = 4t − 3, satisfying the differential equation. Substitution at t = 0 gives 7, satisfying the initial condition. Checking only one would leave the other claim untested.
The expression 2t² − 3t + 9 satisfies the same differential equation but fails the condition. Conversely, many functions pass through the point (0,7) while failing the rate law away from that point.
A solution statement should therefore identify the function, its relevant domain and the conditions it satisfies. In applications, a third check returns to the quantities: whether the result has appropriate units and remains within the model’s intended range.
Continue into: The Core Aim of Mathematics Mastery | Differentiation · The Core Aim of Mathematics Mastery | Integration.
CHAPTER 5 OF 19 · Solve and interpret
5. Use separation while retaining the solutions division can lose
For y′ = 2t y, separation on an interval where y is nonzero gives dy/y = 2t dt. Integration gives ln|y| = t² + C, leading to y = A exp(t²) with nonzero A on that branch.
Before division, examine the excluded possibility. The constant function y = 0 satisfies the original equation. Including A = 0 in the final expression restores it, but that inclusion needs the direct check; it was not justified by dividing by y.
With y(0) = −3, the particular solution is y = −3 exp(t²). Differentiation gives −6t exp(t²), which equals 2t times y. Its negative sign is compatible with the differential equation; the sign of y′ also depends on t.
The same habit matters when dividing by a product such as P(1 − P/K). Inspect the zeros first. A method that produces a useful nonconstant family can still omit the equilibrium solutions unless they are recorded separately.
Continue into: The Core Aim of Mathematics Mastery | Logarithms.
Use a fictional model with temperature T in degrees Celsius and t in minutes. Let the surroundings remain at 25°C, the initial object temperature be 85°C and the rate law be T′ = −0.2(T − 25).
Set θ = T − 25. Because the surrounding temperature is constant, θ′ = T′, and the equation becomes θ′ = −0.2θ. The initial excess is 60°C, giving θ(t) = 60 exp(−0.2t) and T(t) = 25 + 60 exp(−0.2t).
After five minutes, the model predicts approximately 47.1°C. To reach 40°C, solve 15 = 60 exp(−0.2t), giving t = ln(4)/0.2, approximately 6.93 minutes. The target is reached in the model; the surrounding temperature itself is approached asymptotically.
Check both the rate law and the assumptions. If the surroundings change with time, differentiating θ = T − Tₐ(t) introduces the extra term −Tₐ′(t). Reusing the constant-surroundings solution would silently change the problem.
Continue into: The Core Aim of Mathematics Mastery | Exponential Functions.
CHAPTER 7 OF 19 · Solve and interpret
7. Solve an input-and-loss problem with an integrating factor
Consider the fictional amount model M′ = 6 − 0.3M, with M(0) = 5 grams and time in minutes. The input is 6 grams per minute and the proportional loss constant is 0.3 per minute. Rearrange to M′ + 0.3M = 6.
Multiply by exp(0.3t). The left side becomes the derivative of M exp(0.3t), so integration gives M exp(0.3t) = 20 exp(0.3t) + C. Hence M = 20 + C exp(−0.3t).
The initial condition gives C = −15, producing M(t) = 20 − 15 exp(−0.3t). Initially M′ = 4.5 grams per minute. As M approaches 20 grams, the proportional loss approaches the 6-gram-per-minute input.
This is more than an integration exercise. The equilibrium amount comes from balancing input and loss. If the input changes after a specified time, solve the new equation from that time using the amount reached as the next initial condition. A continuous amount need not have a continuous rate across an abrupt input change.
| Check | Calculation | Interpretation |
|---|---|---|
| Initial amount | M(0) = 20 − 15 = 5 grams | The solution satisfies the supplied starting condition. |
| Rate | M′ = 4.5 exp(−0.3t) grams/minute | Differentiation gives the instantaneous change. |
| Equation | 6 − 0.3M = 4.5 exp(−0.3t) | The derivative and model right-hand side agree. |
| Long-time value | M tends to 20 grams | Input and modelled loss approach balance. |
Continue into: How Mathematical Modelling Works | From Reality and Assumptions to Equations, Validation, Prediction and Decisions.
CHAPTER 8 OF 19 · Solve and interpret
8. Read equilibria and direction before solving explicitly
For an autonomous scalar equation y′ = f(y), an equilibrium value y* satisfies f(y*) = 0. The constant function y(t) = y* is then a solution. Nearby signs of f help describe the direction of motion where the equation is defined.
In M′ = 6 − 0.3M, the equilibrium is 20. Below 20 the derivative is positive; above 20 it is negative. The explicit solution confirms movement towards that equilibrium for every real initial amount in the mathematical model.
A direction field records slopes at locations in the independent-variable and solution plane. It can suggest increasing or decreasing behaviour, stationary curves and how initial conditions select different paths. It does not replace a proof of every feature that appears in a rough drawing.
State what kind of stability is being discussed. Returning towards an equilibrium, remaining nearby after a small perturbation and keeping responses bounded are related questions with different definitions. The linked Nonlinear Dynamics and Control Theory roots extend this analysis.
Continue into: The Gold Standard of Nonlinear Dynamics · The Gold Standard of Control Theory.
CHAPTER 9 OF 19 · Examine the assumptions
9. Use a logistic model without turning its parameter into a promise
A fictional population model is P′ = 0.4P(1 − P/100), with t in days and P(0) = 20. The rate constant is 0.4 per day and the model’s carrying-capacity parameter is 100.
The solution for this initial value is P(t) = 100/[1 + 4 exp(−0.4t)]. The denominator is positive for t ≥ 0. The population rises from 20 towards 100 in this model, while its per-capita growth rate declines as P rises.
The fastest absolute growth along this trajectory occurs at P = 50, because P(1 − P/100) is a downward-opening quadratic with its maximum there. Solving 50 = 100/[1 + 4 exp(−0.4t)] gives t = ln(4)/0.4, approximately 3.47 days.
The constant solutions P = 0 and P = 100 need separate attention if separation divides by their defining factors. A real habitat does not acquire a permanently fixed capacity because a model includes K. Evidence, time scale and environmental change determine whether that simplification is useful.
Continue into: How Science Works | Mathematical Biology — Equations, Networks, Populations, Dynamics and Models of Living Systems.
CHAPTER 10 OF 19 · Examine the assumptions
10. Separate local existence from uniqueness and long-term validity
An initial condition does not automatically guarantee one solution for all future time. Existence asks whether a solution is available. Uniqueness asks whether another solution can satisfy the same problem. Global continuation asks how far a solution can be extended.
For y′ = f(t,y), continuity of f together with appropriate local Lipschitz behaviour in y provides a standard route to local existence and uniqueness. These are sufficient hypotheses whose domain must be checked, not words that can be attached to any equation.
A simple illustration is y′ = y² with y(0) = 1. Its solution is y = 1/(1 − t), on the maximal interval containing zero with t < 1. It becomes unbounded as t approaches 1 from below, despite the smooth right-hand side.
The model therefore has a unique local trajectory and a finite-time limitation. A solver producing a line beyond the singularity has not extended that same initial-value solution through an ordinary finite value. Domain information belongs in the answer.
Continue into: How Mathematics Works | Differential Equations.
CHAPTER 11 OF 19 · Examine the assumptions
11. See why one starting point can allow several trajectories
Consider y′ = 2√y for nonnegative y, with y(0) = 0 and t ≥ 0. The zero function is a solution. So is y = t² on this nonnegative time domain.
There are also delayed-start solutions: choose any c ≥ 0, set y(t) = 0 for 0 ≤ t ≤ c and y(t) = (t − c)² for t ≥ c. At t = c the function and derivative match, and substitution verifies the equation on both sides and at the join.
These different trajectories share the same initial condition. The right-hand side lacks the needed local Lipschitz behaviour at y = 0, so a familiar uniqueness theorem cannot be invoked there.
This example is a mathematical diagnosis, not a claim that a physical object can spontaneously choose an unexplained start time. If a proposed model leaves future evolution nonunique, ask whether additional physical structure, a revised equation or a different interpretation is needed.
Continue into: The Gold Standard of Mathematical Modelling.
CHAPTER 12 OF 19 · Follow motion and computation
12. Give second-order motion enough initial information
The fictional normalised equation x″ + 2x′ + 4x = 0 describes a damped oscillator. To specify its motion, supply x(0) = 1 and x′(0) = 0. An initial position alone would leave multiple possible initial velocities.
The characteristic equation λ² + 2λ + 4 = 0 has roots −1 ± i√3. The real solution family is x(t) = exp(−t)[A cos(√3t) + B sin(√3t)].
The position condition gives A = 1. Differentiating and evaluating at zero gives −A + √3B = 0, so B = 1/√3. The result has an exponentially decreasing oscillatory envelope.
The characteristic method applies here because the equation is linear with constant coefficients. It does not turn every nonlinear or variable-coefficient second-order equation into the same polynomial problem. Structural Vibration, Dynamics and the Eigenvalues branch explain the wider connections.
Continue into: Why Mathematics? | Structural Vibration, Natural Frequency and Damping · The Core Aim of Mathematics Mastery | Complex Numbers.
CHAPTER 13 OF 19 · Follow motion and computation
13. Turn a higher-order equation into a system of states
For the oscillator, define v = x′. The second-order equation becomes two first-order equations: x′ = v and v′ = −4x − 2v. The state is the pair (x,v), and the initial state is (1,0).
A graph of x against t shows one component. A phase-plane trajectory plots the evolving pair, so it represents position and velocity together. Two instants with the same position can have different velocities and different immediate futures.
In matrix form the system is z′ = Az, where z contains x and v and A has rows (0,1) and (−4,−2). Its eigenvalues recover the characteristic roots of the earlier equation.
State-space reasoning becomes especially useful for interacting quantities. The physical meaning of each state still matters, including its units and constraints. A matrix makes the relationships compact; it does not remove the need to explain why those relationships model the system.
Continue into: The Core Aim of Mathematics Mastery | Eigenvalues and Eigenvectors · What Is a Trajectory? | How a System’s History Becomes a Path Through State Space.
CHAPTER 14 OF 19 · Follow motion and computation
14. Treat boundary conditions as a different problem structure
For y″ = 0 on 0 ≤ x ≤ 3, integration gives y = Ax + B. The conditions y(0) = 2 and y(3) = 8 give B = 2 and A = 2, yielding y = 2x + 2.
The conditions are at two different locations. This is a boundary-value problem, rather than the initial-value problem that would specify y and y′ together at one starting location.
Boundary conditions do not automatically select a unique solution. For y″ + y = 0 on [0,π], the conditions y(0) = 0 and y(π) = 0 are satisfied by every function y = A sin x, for any real A.
The equation is regular and linear, yet this boundary-value problem has infinitely many solutions. Check the compatibility and independence of the conditions for the actual problem. The distinction helps explain why initial-value intuition cannot be transferred unchanged to beams, spatial fields and eigenvalue problems.
Continue into: Why Mathematics? | Finite-Element Analysis, Mesh Convergence and Stress Concentrations.
CHAPTER 15 OF 19 · Follow motion and computation
15. Compute a numerical step and identify what it approximates
Euler’s method approximates y′ = f(t,y) using yₙ₊₁ = yₙ + h f(tₙ,yₙ), with tₙ₊₁ = tₙ + h. It follows the slope at the beginning of each step, rather than the full curvature across the interval.
Take the normalised test problem y′ = −4y, y(0) = 1. With h = 0.1, one step gives y₁ = 1 + 0.1(−4) = 0.6. The exact value at t = 0.1 is exp(−0.4), approximately 0.6703.
Repeating the update gives yₙ = (1 − 4h)ⁿ. At t = 1, h = 0.1 gives approximately 0.00605; h = 0.05 gives approximately 0.01153. The exact value is exp(−4), approximately 0.01832.
The calculation can be reproduced, but reproducibility alone does not establish sufficient accuracy. State the equation, method, starting value, step size and target time so another reader can understand what was approximated.
Continue into: The Core Aim of Mathematics Mastery | Numerical Methods.
CHAPTER 16 OF 19 · Follow motion and computation
16. Keep numerical stability separate from physical behaviour
For the same Euler calculation, h = 1 gives the multiplier 1 − 4h = −3. The numerical values are 1, −3, 9 and −27, although the true solution exp(−4t) is positive and decays.
The growing oscillation comes from the numerical method and step size, not from instability in the underlying differential equation. For this scalar test, errors shrink under the Euler update when |1 − 4h| < 1, giving 0 < h < 0.5.
Stability does not by itself guarantee adequate accuracy. The h = 0.1 result is stable yet noticeably below the exact value at t = 1. Method order, tolerances, time scale and accumulated error all matter.
Runge–Kutta methods, embedded error estimates and implicit methods provide further tools with their own properties. A stiff problem can force an explicit method to use small steps for stability even when the solution of interest changes slowly. Choose a method for the problem instead of trusting a smooth plotted curve.
| Euler step h | Multiplier 1 − 4h | What the update does |
|---|---|---|
| 0.1 | 0.6 | Positive decay; accuracy still needs assessment. |
| 0.25 | 0 | Jumps to zero after one step; not the exact exponential curve. |
| 0.5 | −1 | Alternates without decay from a nonzero start. |
| 1 | −3 | Alternates with growing magnitude. |
Continue into: Why Mathematics? | Runge–Kutta Methods, Intermediate Slopes, Step Size and Dynamic Simulation · Why Mathematics? | Runge–Kutta–Fehlberg Embedded Estimates, Local Error and Adaptive Step Size.
CHAPTER 17 OF 19 · Transfer and evaluate
17. Extend the account to space-dependent equations
When temperature varies along a rod, a single T(t) may be insufficient. An ideal one-dimensional diffusion model uses T(x,t), with ∂T/∂t = α∂²T/∂x² for constant diffusivity α under its stated assumptions.
The derivative on the left describes change in time at a location. The spatial second derivative on the right describes how the local temperature profile curves. Initial temperature and appropriate boundary conditions complete the problem specification.
A spatial discretisation can turn such a partial differential equation into many coupled ordinary differential equations. A time-stepping method then introduces another approximation. Refining the time step while leaving an inadequate spatial mesh does not test every source of error.
Diffusion and wave equations describe different behaviours. The existence of a derivative with respect to time does not make all spatial models interchangeable. Crank–Nicolson, Finite-Element Analysis, weather modelling and the wider numerical routes deepen particular methods and applications.
Continue into: Why Mathematics? | Crank–Nicolson Time Stepping, Diffusion Equations and Implicit Averages · Why Mathematics? | Weather Forecasting, Differential Equations and Numerical Models.
CHAPTER 18 OF 19 · Transfer and evaluate
18. Separate fitted parameters from evidence about the model
A cooling curve can be fitted by adjusting k, but a fit does not establish every assumption behind the rate law. Different parameter choices, measurement bias or omitted processes can produce similar observations over a short interval.
Begin with the question the model needs to answer. Record how parameters were estimated, what observations were used and which conditions remained fixed. Where possible, assess predictions against relevant observations that did not determine the fit.
Test sensitivity. If a small change in a plausible parameter or initial value produces a large change in the quantity being reported, communicate that dependence. Additional decimal places in a fitted constant do not remove structural uncertainty.
Mathematical correctness and empirical usefulness are separate checks. A solution can satisfy its equation perfectly while the equation is unsuitable for the observed system. The Mathematical Modelling and Digital Twins guides provide the next routes into that distinction.
Continue into: The Gold Standard of Mathematical Modelling · Why Mathematics? | Digital Twins, State Estimation and Parameter Calibration.
CHAPTER 19 OF 19 · Transfer and evaluate
19. Diagnose the first mistaken move and test transfer
A learner can separate variables but lose an equilibrium through division. Another can integrate correctly but apply a condition to the wrong function. A third can produce accurate code for an unstable numerical scheme. These need different repairs.
Ask the student to identify the unknown function, rate law, units and conditions before solving. Then require a differentiation check, a domain statement and an interpretation of the sign or limiting behaviour.
For a transfer task, change one condition in a worked example. Replace constant ambient temperature with a time-varying one, change the input after five minutes or halve a numerical step. The learner should explain which reasoning survives and which part must be rebuilt.
The endpoint is independent judgement. A strong answer states what was solved, how it was checked and what remains uncertain. Use the connected directory to repair the specific gap rather than adding unrelated difficulty to a method that is not yet understood.
Continue into: The Core Aim of Mathematics Mastery | Differential Equations · The Gold Standard of Education | Learning That Transfers into Life.
Contents · Previous chapter · Continue to the article directory
Connected specialist article directory
Each route below has a different job. Use the subject guides to understand the structure, the mathematical branches to repair a dependency and the specialist methods to answer a narrower computational or application question.
Begin with the subject and its mathematical dependencies
| Existing article | Use this route to |
|---|---|
| How Mathematics Works | Differential Equations | Follow the existing subject mechanism, from rate laws to conditions, solutions and stability. |
| The Core Aim of Mathematics Mastery | Differential Equations | Build the established learning sequence and repair common procedural errors. |
| The Gold Standard of Calculus | Place derivatives and integrals inside the wider subject. |
| The Core Aim of Mathematics Mastery | Differentiation | Check the rate of change of a proposed solution. |
| The Core Aim of Mathematics Mastery | Integration | Recover functions while retaining constants and conditions. |
| The Core Aim of Mathematics Mastery | Exponential Functions | Understand the response generated by proportional rate laws. |
| The Core Aim of Mathematics Mastery | Logarithms | Solve for times and interpret logarithms in separated equations. |
| The Core Aim of Mathematics Mastery | Complex Numbers | Connect characteristic roots with real oscillatory solutions. |
Check models, states and mathematical structure
Choose and evaluate numerical methods
Connect applications to the appropriate model
Return to learning and the wider collection
| Existing article | Use this route to |
|---|---|
| The Gold Standard of Education | Learning That Transfers into Life | Judge learning by whether the reasoning survives a changed condition. |
| How X Works | Explore the wider mechanism collection and its connected subjects. |
A practical quality check
Before accepting a solution, identify the unknown function and independent variables, the equation’s domain, the initial or boundary data and the method’s conditions. A differentiable expression on the wrong interval is not a complete answer to the stated problem.
Check the solution in the original equation, not only in a rearranged form that involved division or another restriction. Then check every supplied condition. Return any excluded equilibrium or exceptional case to the original relationship so it is neither lost nor added without justification.
For a numerical answer, distinguish discretisation error from input uncertainty and model inadequacy. Reducing a step may help the approximation while leaving a biased measurement or an omitted mechanism untouched. A dependable account names the error source being investigated and the evidence that tests it.
Questions readers ask
Is solving a differential equation the same as integrating?
Integration is central to many methods, but a differential equation can contain the unknown function inside its rate law. It may require transformation, a system of equations, qualitative analysis or numerical approximation. The method must fit the equation and its conditions.
Does an initial condition always select one solution?
No. Appropriate existence and uniqueness hypotheses need to be satisfied. Even a unique local solution can fail to extend for all future time. The nonunique square-root example and the finite-time blow-up example show different reasons to retain those distinctions.
Why do equilibrium solutions disappear in some workings?
Division by a factor that is zero at an equilibrium excludes that case from the calculation. Check the factor’s zeros before division and test possible constant solutions directly in the original equation.
Is a stable numerical answer necessarily accurate?
No. Stability concerns how perturbations or numerical modes behave under specified conditions. Accuracy concerns agreement with the solution to the required level. The stable Euler example still has appreciable error at the stated target time.
Do more boundary conditions always help?
They must be appropriate and compatible, rather than merely numerous. They can conflict or fail to be independent. The example y″ + y = 0 with zero values at both ends of [0,π] admits infinitely many solutions.
What is a good way to check genuine understanding?
Ask the learner to derive and verify one solution, then change an assumption. A variable ambient temperature, a switched input or a different numerical step reveals whether the learner understands the relationships or is repeating the original procedure.
Primary references and further learning
OpenStax · Basics of Differential Equations — Definitions, order, initial-value problems and verification of solutions.
OpenStax · Direction Fields and Numerical Methods — Qualitative slopes and introductory numerical approximation.
OpenStax · Separable Equations — Separation, conditions and rate-law applications.
OpenStax · The Logistic Equation — An introductory route into logistic growth and its interpretation.
MIT OpenCourseWare · Differential Equations topic notes — Further linear, nonlinear, system, numerical and partial-differential-equation study.
MIT OpenCourseWare · Honors Differential Equations — A route into existence, uniqueness, linear systems, transforms and wider theory.
The polynomial, cooling, input-and-loss, logistic, square-root, oscillator, boundary-value and Euler cases are original teaching examples. Their numerical values illustrate the supplied assumptions. The references provide further learning and checking routes; none turns a fictional parameter into a measured physical fact.
Continue into The Gold Standard of Education | Learning That Transfers into Life or the Differential Equations mechanism guide. For the next changing-system question, write down the quantity, rate law and conditions first, then decide which part can be solved and which part still needs evidence.