Verifying Solutions

A differential equation (DE) is an equation that relates a function to its derivatives. When we build mathematical models of real-world phenomena — population growth, cooling coffee, charging capacitors — we write down a DE whose solution describes how the system evolves. But how do we know whether a proposed function actually satisfies that equation? The process of checking is called verifying a solution, and it is one of the most fundamental skills in differential equations. Verifying a solution means substituting the candidate function and its required derivatives directly into the differential equation and confirming that both sides are equal for all values of the independent variable. If the equation holds as an identity, the function is a solution; if even one value fails, it is not. This sounds simple, but careful algebra and correct differentiation are essential — one sign error can lead to a wrong conclusion. Verification is important for several reasons. First, it builds confidence that an answer obtained by solving is correct. Second, it lets us test functions that someone else proposes without knowing how they were derived. Third, it deepens understanding of what a solution really means: not just a formula, but a function whose rate-of-change behaviour matches the rule encoded in the differential equation. In this article you will learn the step-by-step verification procedure and apply it to several examples.

Key takeaways
  • Identify the differential equation and the candidate solution
  • Compute all required derivatives
  • Substitute into the differential equation
  • Simplify both sides and check for identity
  • Check initial or boundary conditions (if given)

Identify the differential equation and the candidate solution

Write down the DE clearly, noting its order (the highest derivative present) and which variable is independent (usually x or t) and which is dependent (usually y). Write down the proposed solution y = f(x) that you want to test.

Compute all required derivatives

Determine which derivatives appear in the DE. If it is a first-order DE, compute dy/dx. If it is second-order, compute dy/dx and d²y/dx². Simplify each derivative fully before substituting.

Substitute into the differential equation

Replace y and every derivative in the DE with the expressions you computed. Work on each side of the equation separately. Do not assume equality yet — that is what you are trying to establish.

Simplify both sides and check for identity

Use algebra, trigonometric identities, or exponential rules to simplify both sides until they are in the same form. If both sides reduce to the same expression for all values of the independent variable, the function is a valid solution. State your conclusion explicitly.

Check initial or boundary conditions (if given)

A general solution contains arbitrary constants (like C). An initial condition such as y(0) = 3 pins down the value of C. Substitute the initial condition into the proposed particular solution to verify it satisfies those constraints as well. A function can satisfy the DE but fail an initial condition, in which case it is not the correct particular solution.

The Key Equation

dydx=ky\frac{dy}{dx} = ky
  • y — Dependent variable (varies (e.g. population, mass)): The unknown function we seek; its value changes with x.
  • \frac{dy}{dx} — First derivative (same as y per unit x): The instantaneous rate of change of y with respect to x.
  • k — Growth/decay constant (per unit x (e.g. per second)): A real constant; positive k gives growth, negative k gives decay.
  • e^{kx} — Exponential function (dimensionless multiplier): The base of the natural exponential; central to solutions of linear DEs with constant coefficients.

Worked Example

Problem. Verify that y = 3e^{2x} is a solution of the differential equation dy/dx − 2y = 0, and confirm it satisfies the initial condition y(0) = 3.

y=3e2x satisfies dydx−2y=0 with y(0)=3✓y = 3e^{2x} \text{ satisfies } \frac{dy}{dx} - 2y = 0 \text{ with } y(0)=3 \checkmark

Answer. y = 3e^{2x} is a valid particular solution of dy/dx − 2y = 0 satisfying y(0) = 3.

Figure
Diagram
Verifying y = 3e²ˣ in dy/dx − 2y = 0xy00.20.40.60.81.0y=3e²ˣ(0, 3)ΔxΔyslope = dy/dx = 6e²ˣ= 2 × (3e²ˣ) = 2y ✓3At every point on the curve, slope dy/dx equals 2y — confirming the DE holds identically.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus alongside Newton and invented the dy/dx notation we still use today. His systematic notation made it natural to write differential equations and manipulate derivatives as algebraic objects, laying the groundwork for the entire theory of DEs and the modern procedure of verifying solutions by substitution. Leibniz designed one of the first mechanical calculators — the Stepped Reckoner — that could multiply and divide, not just add. He built it to save mathematicians from 'slavish' arithmetic, so they could focus on deeper reasoning — much like verifying solutions by logic rather than brute calculation.

  • Separable Differential Equations — Separation of variables is the most common technique for solving first-order DEs; once you obtain a solution this way, verification is the next immediate step.
  • General and Particular Solutions — Verification must address both the DE itself and any initial conditions; understanding the difference between general and particular solutions clarifies which conditions to check.
  • The Chain Rule — Computing derivatives of composite functions like e^{kx} or sin(ωx) during verification requires confident use of the chain rule.
  • Exponential Growth and Decay Models — Exponential functions are the most frequently encountered solutions in modelling DEs; verifying that y = Ce^{kt} solves dy/dt = ky is a canonical application of this concept.