Section 4.3 Use Laplace transformation to solve the differential equations

Objective:

1. Use Laplace transformation to solve basic differential equations.

In this section, we are solving more IVP using Laplace. 

Example 1: Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }-3y=e^{2t}$, $y(0)=-1$, $y^{\prime }(0)=2$.

Exercise 1: Use Laplace Transform to solve the IVP. $y^{″}+y^{\prime }-6y=e^{3t}$, $y(0)=-2$, $y^{\prime }(0)=1$.

Example 2: Use Laplace Transform to solve the IVP. $y^{″}+y^{\prime }+y=\text{cos}(t)$, $y(0)=1$, $y^{\prime }(0)=0$.

Exercise 2: Use Laplace Transform to solve the IVP. $y^{″}+2y^{\prime }+5y=\text{sin}(t)$, $y(0)=0$, $y^{\prime }(0)=1$.

Corollary: Suppose that $f$ is a function for which the following hold:

(1) $f$ is continuous and $f^{\prime }$ is piecewise continuous on $[0,b]$ for all $b>0$.

(2) $|f(t)|\le K{e}^{at}$, when $t\ge M$,for constants $a,K,M,$ with $K,M>0$.

(3) $F(s)=L\{f(t)\}$.

Then the Laplace Transform of $(-t{)}^{n}f(t)$ exists for $s>a$, with

$$L\{(-t{)}^{n}f(t)\}=F^{(n)}(s).$$

Proof for $n=1$ case:

Example 3: Find Laplace Transform of $t{e}^{at}$.

Exercise 3: Find Laplace Transform of $t\text{sin}(2t)$.

Group Work

1. Use Laplace Transform to solve the IVP. $y^{″}-4y^{\prime }+13y=t$, $y(0)=0$, $y^{\prime }(0)=1$. 

2. Find Laplace Transform of $t^2\text{cos}(bt)$.

3. Find Laplace Transform of $t{e}^{at}\text{sin}(bt)$.