7.2 Verifying Solutions

A solution is a function

A solution to a differential equation is a function (not a number) that makes the equation true when substituted in.

How to verify

Substitute the proposed function and its derivative into the equation and check that both sides are equal. Example: to check y = e^(2x) solves y' = 2y, compute y' = 2e^(2x), which equals 2y. ✓

Key terms

Solution

A function that satisfies the differential equation.

Verify

Substitute the function and its derivative.

Check

Both sides must be equal.

Not a solution

If substitution fails to satisfy the equation.

Example

y = e^(2x) solves y' = 2y since y' = 2e^(2x).

Try an AP-style question

Which function is a solution to y' = 2y?

1. y = x² + 5
2. y = e^(x²)
3. y = 5e^(2x)
4. y = 2eˣ