Solving IVP with Laplace

For

$$\begin{array}{rl}& a_n{y}^{(n)}+a_{n-1}{y}^{(n-1)}+\cdots +a_1{y}^{\prime }+a_{0}y=f(t)\\ & \\ & y(0)=y_0,y^{\prime }(0)=y_1,\dots ,y^{(n-1)}(0)=y_{n-1}\end{array}$$

Steps

1. Take the Laplace Transform for both sides of the equation
2. Solve for $Y(s)=\mathcal{L}\{y(t)\}$
3. $y(t)={\mathcal{L}}^{-1}\{Y(s)\}$

When Initial Conditions at y(a)

1. Substitute $y(t)=w(t+a)$
2. Solve Normally
3. Substitute $w(t)=y(t-a)$

Notes

• $\mathcal{L}\{y^{\prime }(t)\}=sY(s)-y(0)$
• $\mathcal{L}\{y^{\prime \prime }(t)\}=s^{2}Y(s)-sy(0)-y^{\prime }(0)$