AP Calculus BC / Unit 7: Differential Equations
7.2 Verifying Solutions for Differential Equations
Notes
To check a solution, find its derivative(s) and substitute into the equation. If both sides match, it's a solution.
- A general solution has $+C$; a particular solution uses an initial condition.
Practice
Is $y=e^{3x}$ a solution of $y'=3y$?
$y'=3e^{3x}=3y$.
Which is a solution of $y''+y=0$?
$(\sin x)''=-\sin x$.
$y=Ce^{2x}$ is a general solution of $y'=2y$. With $y(0)=5$, $C=$
$Ce^0=5$.
Ready for more? Take the Unit 7 test.