4.4 Solving Initial Value Problems

Method of Laplace Transform for IVP

General Approach:

1. Apply the Laplace Transform to each term of the differential equation. Use the properties of the Laplace Transform listed in Tables 4.1 and 4.2 to obtain an equation in terms of ${\displaystyle Y\left(s\right)}$. The Laplace Transform of the derivatives are

 ${\displaystyle L\{f\prime \left(t\right)\}=sF\left(s\right)-f\left(0\right)}$

 ${\displaystyle L\{f\prime \prime \left(t\right)\}=s^{2}F\left(s\right)-sf\left(0\right)-f\prime \left(0\right)}$

2. The transforms of derivatives involve initial conditions at ${\displaystyle t=0}$. Apply the initial conditions.

3. Simplify the transformed equation to isolate ${\displaystyle Y\left(s\right)}$.

4. If needed, use partial fraction decomposition to break down ${\displaystyle Y\left(s\right)}$ into simpler components.

5. Determine the inverse Laplace Transform using the tables and linearity property to find ${\displaystyle y\left(t\right)}$.

Shortcut Approach:

1. Find the characteristic polynomial of the differential equation ${\displaystyle p\left(s\right)=a{s}^2+bs+c}$.

2. Substitute ${\displaystyle p\left(s\right)}$, ${\displaystyle F\left(s\right)=L\left\{f\left(t\right)\right\}}$, and the initial conditions into the equation

 ${\displaystyle Y\left(s\right)=\frac{F\left(s\right)+a(y\prime \left(0\right)+sy\left(0\right))+by\left(0\right)}{p\left(s\right)}}$ (4.4.1)

3. If needed, use partial fraction decomposition to break down ${\displaystyle Y\left(s\right)}$ into simpler components.

4. Determine the inverse Laplace transform of ${\displaystyle Y\left(s\right)}$ using the tables and linearity property to find ${\displaystyle y\left(t\right)}$.

Solve the initial value problem.

 ${\displaystyle y\prime \prime -5y\prime +6y=4e^{-2t} ;}$    ${\displaystyle y\left(0\right)=-1, y\prime \left(0\right)=2}$

Solve the initial value problem.

 ${\displaystyle y\prime \prime +4y=3\sin \left(t\right) ;}$    ${\displaystyle y\left(0\right)=1, y\prime \left(0\right)=-1}$

1. Solve the IVP by using the inverse Laplace Transform ${\displaystyle y\left(t\right)=L^{-1}\left\{Y\left(s\right)\right\}}$

    ${\displaystyle y\prime \prime +3y\prime -10y=0, y\left(0\right)=-1, y\prime \left(0\right)=2}$

   Show/Hide Answer

   ${\displaystyle y\left(t\right)=-\frac{3}{7}{e}^{2t}-\frac{4}{7}{e}^{-5t}}$

2. Solve the IVP by using the inverse Laplace Transform ${\displaystyle y\left(t\right)=L^{-1}\left\{Y\left(s\right)\right\}}$

    ${\displaystyle y\prime \prime +6y\prime +13y=0, y\left(0\right)=2, y\prime \left(0\right)=0}$

   Show/Hide Answer

   ${\displaystyle y\left(t\right)=e^{-3t}(2\cos \left(2t\right)+3\sin \left(2t\right))}$

3. Solve the IVP by using the inverse Laplace Transform ${\displaystyle y\left(t\right)=L^{-1}\left\{Y\left(s\right)\right\}}$

    ${\displaystyle y\prime \prime -8y\prime +16y=0, y\left(0\right)=1, y\prime \left(0\right)=-1}$

   Show/Hide Answer

    ${\displaystyle y\left(t\right)=e^{4t}(1-5t)}$