Section 4.3 Use Laplace transformation to solve the differential equations

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

Example 1: Use Laplace Transform to solve the IVP. [latex]y''+2y'-3y=e^{2t}[/latex], [latex]y(0)=-1[/latex], [latex]y'(0)=2[/latex].

Exercise 1: Use Laplace Transform to solve the IVP. [latex]y''+y'-6y=e^{3t}[/latex], [latex]y(0)=-2[/latex], [latex]y'(0)=1[/latex].

Example 2: Use Laplace Transform to solve the IVP. [latex]y''+y'+y=\text{cos}(t)[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=0[/latex].

Exercise 2: Use Laplace Transform to solve the IVP. [latex]y''+2y'+5y=\text{sin}(t)[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=1[/latex].

Corollary: Suppose that [latex]f[/latex] is a function for which the following hold:

(1) [latex]f[/latex] is continuous and [latex]f'[/latex] is piecewise continuous on [latex][0,b][/latex] for all [latex]b \gt 0[/latex].

(2) [latex]|f(t)|\leq Ke^{at}[/latex], when [latex]t\geq M[/latex],for constants [latex]a,K,M,[/latex] with [latex]K,M \gt 0[/latex].

(3) [latex]F(s)=L\{f(t)\}[/latex].

Then the Laplace Transform of [latex](-t)^{n}f(t)[/latex] exists for [latex]s \gt a[/latex], with

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

Proof for [latex]n=1[/latex] case:

Example 3: Find Laplace Transform of [latex]te^{at}[/latex].

Exercise 3: Find Laplace Transform of [latex]t\text{sin}(2t)[/latex].

Group Work

1. Use Laplace Transform to solve the IVP. [latex]y''-4y'+13y=t[/latex], [latex]y(0)=0[/latex], [latex]y'(0)=1[/latex].

2. Find Laplace Transform of [latex]t^{2}\text{cos}(bt)[/latex].

3. Find Laplace Transform of [latex]te^{at}\text{sin}(bt)[/latex].

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