Verifying Solutions for Differential Equations
Introduced| English | Meaning |
|---|---|
| solution/səˈluːʃn/ | A value, set of values or function that satisfies an equation, inequality or other mathematical condition. |
Is this function actually a solution?
- Before solving a differential equation, you should be able to check a proposed answer.
- A solution 解 of a differential equation is a function that makes the equation true.
- The test is purely mechanical: differentiate the candidate and substitute.
- If both sides match for all $x$, it's a solution; if not, it isn't.
The verification recipe
- 1. Take the candidate function and compute its derivative(s).
- 2. Substitute the function and its derivatives into the differential equation.
- 3. Simplify both sides and check that they are equal (as identities in $x$).
- No integration needed — verifying is just differentiating and plugging in.
The constant gives a family of curves; an initial condition picks out one
To verify a candidate solution, you...
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Differentiate the candidate, substitute, check both sides match.
A worked check
- Does $y=e^{3x}$ solve $\dfrac{dy}{dx}=3y$?
- Differentiate: $\dfrac{dy}{dx}=3e^{3x}$.
- Substitute into the right side: $3y=3e^{3x}$.
- Left $=$ right ($3e^{3x}=3e^{3x}$), so yes, $y=e^{3x}$ is a solution. ✓
Solutions of dy/dx = ky
Each exponential $y=Ce^{kx}$ threads the slope field — verifying one just confirms its derivative matches $ky$.
$y=e^{3x}$ is a solution of $\dfrac{dy}{dx}=3y$.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$\frac{dy}{dx}=3e^{3x}=3y$. ✓
Both sides must be equal for ____ values of $x$, not just one, to confirm a solution.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
A solution satisfies the equation identically.
General vs. particular solutions
- $y=Ce^{3x}$ (with any constant $C$) also solves $\dfrac{dy}{dx}=3y$ — the whole family is the general solution.
- Pick a specific $C$ and you get a particular solution.
- Verifying works the same for either: differentiate and check the equation holds.
- The constant carries through the differentiation, so the check still balances.
Verifying a given solution requires integrating the differential equation.
Difficulty 1/5 · 100 coins + a small bonus on first completion today
Verifying is differentiate-and-substitute; no integration.
For which constants $C$ does $y=Ce^{3x}$ solve $\dfrac{dy}{dx}=3y$?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$\frac{dy}{dx}=3Ce^{3x}=3y$ for all $C$ — the general solution.
Is $y=x^2+C$ a solution of $\dfrac{dy}{dx}=2x$?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
$\frac{dy}{dx}=2x$ regardless of $C$.
Verifying is not solving — you're only checking a given candidate, so no integration is involved. Substitute the function and its derivative into the equation and confirm both sides are identical for all $x$, not just at one point. A match at a single $x$ isn't enough.
Verify that $y=x^2+C$ solves $\dfrac{dy}{dx}=2x$.
- Differentiate: $\dfrac{dy}{dx}=2x$ (the $+C$ vanishes).
- The differential equation says $\dfrac{dy}{dx}=2x$.
- Both sides equal $2x$, so $y=x^2+C$ is a solution for every constant $C$. ✓
To verify a solution of a differential equation, differentiate the candidate and substitute it (with its derivatives) into the equation; it's a solution iff both sides are equal for all $x$. No integration required — verifying is differentiate-and-check. A family with a constant $C$ is the general solution.
Write from a check that needs no integral
- A candidate is y = 3 e to the 2x.
- The equation is dy/dx = 2y.
Is the candidate a solution?
6 e to the 2x equals 2 times 3 e to the 2x.
y = 3 e to the 2x and dy/dx = 2y. The candidate is what?
Difficulty 1/5 · 100 coins + a small bonus on first completion today
The derivative matches 2y everywhere.