Worked exercise — y'' + 9y = 0 with initial conditions

Problem

Solve $y''+9y=0$ with $y(0)=1$, $y'(0)=0$.

Solution

Characteristic equation $\lambda^2+9=0$, roots $\lambda=\pm 3i$ ($\alpha=0$, $\beta=3$). General solution: $y(x) = C_1\cos(3x) + C_2\sin(3x).$ From $y(0)=1$: $C_1=1$. Since $y'(x)=-3C_1\sin(3x)+3C_2\cos(3x)$, from $y'(0)=3C_2=0$ it follows that $C_2=0$. Hence: $\boxed{y(x) = \cos(3x)}$

Links

Topics: Differential equations
Concepts: Initial condition · Second-order linear ODE · Characteristic equation · Harmonic oscillator
Functions: Cosine
Skills: Solving equations
Exercise type: Equation solving