Section 2.8. Solutions of non-homogeneous differential equations with repeated undetermined
Objective:
1. Setting up a particular solution for repeated undetermined.
2. Find the solution of a particular solution for repeated undetermined.
In previous section, we learn to find general solution of [latex]y''+p(t)y'+q(t)y=g(t)[/latex]. All examples that we have practiced have the general solution of [latex]y''+p(t)y'+q(t)y=0[/latex] is different to the presentation of [latex]g(t)[/latex]. At this section, we will work on the case that the general solutions of [latex]y''+p(t)y'+q(t)y=0[/latex] have common terms with [latex]g(t)[/latex].
Example 1: Find a general solution of [latex]y''+2y'-3y=4e^{-3t}[/latex].
Exercise 1: Find a general solution of [latex]y''+5y'+6y=e^{-2t}[/latex].
Example 2: Find a general solution of [latex]y''+6y'+8y=te^{-4t}[/latex].
Exercise 2: Find a general solution of [latex]y''+y'-6y=te^{2t}[/latex].
Example 3: Find the solution of [latex]y''+9y=4\text{cos}(3t)[/latex], [latex]y(0)=3[/latex], and [latex]y'(0)=-2.[/latex]
Exercise 3: Find a general solution of [latex]y''+25y=\text{sin}(5t)[/latex], [latex]y(0)=-1[/latex], and [latex]y'(0)=5.[/latex]
Example 4: Find a general solution of [latex]y''+4y'+4y=te^{-2t}+5[/latex].
Exercise 4: Find a general solutions of [latex]y''-6y'+9y=te^{3t}+3[/latex].
Summary: The general solution of [latex]y''+p(t)y'+q(t)y=g(t)[/latex] is the sum of the complementary solution, [latex]y_{c}[/latex] of the ODE and the particular solution [latex]y_{p}=t^{s}Y(t)[/latex] where [latex]Y(t)[/latex] is depending on the presentation of [latex]g(t)[/latex] and [latex]s[/latex] is the smallest positive integer such that [latex]y_{p}[/latex] and [latex]y_{c}[/latex] have distinct presentations.
Group Work: Find [latex]y_{p}[/latex] presentation. Do not solve for the undetermined.
1. [latex]y''+4y'=3t^{3}+t^{2}e^{-4t}+\text{sin}(2t)[/latex]
2. [latex]y''+4y=3t^{3}+t^{2}e^{-4t}+\text{sin}(2t)[/latex]
3. [latex]y''+4y'+4y=3t^{3}+t^{3}e^{-2t}+\text{sin}(2t)[/latex]
4. [latex]y''+2y'+2y=3t^{3}+t^{2}e^{-4t}+t\text{cos}(t)[/latex]
5. [latex]y''+2y'+5y=3t^{3}+t^{2}e^{-4t}+e^{-t}\text{sin}(2t)[/latex]