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 $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$. All examples that we have practiced have the general solution of $y^{″}+p(t)y^{\prime }+q(t)y=0$ is different to the presentation of $g(t)$. At this section, we will work on the case that the general solutions of $y^{″}+p(t)y^{\prime }+q(t)y=0$ have common terms with $g(t)$. 

Example 1: Find a general solution of $y^{″}+2y^{\prime }-3y=4e^{-3t}$. 

Exercise 1: Find a general solution of $y^{″}+5y^{\prime }+6y=e^{-2t}$.

Example 2: Find a general solution of $y^{″}+6y^{\prime }+8y=t{e}^{-4t}$. 

Exercise 2: Find a general solution of $y^{″}+y^{\prime }-6y=t{e}^{2t}$.

Example 3:  Find the solution of $y^{″}+9y=4\text{cos}(3t)$, $y(0)=3$, and $y^{\prime }(0)=-2.$ 

Exercise 3: Find a general solution of $y^{″}+25y=\text{sin}(5t)$, $y(0)=-1$, and $y^{\prime }(0)=5.$ 

Example 4:  Find a general solution of $y^{″}+4y^{\prime }+4y=t{e}^{-2t}+5$. 

Exercise 4: Find a general solutions of $y^{″}-6y^{\prime }+9y=t{e}^{3t}+3$.

Summary: The general solution of $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$ is the sum of the complementary solution, $y_c$ of the ODE and the particular solution $y_p=t^{s}Y(t)$ where $Y(t)$ is depending on the presentation of $g(t)$ and $s$ is the smallest positive integer such that $y_p$ and $y_c$ have distinct presentations. 

Group Work: Find $y_p$ presentation. Do not solve for the undetermined. 

1. $y^{″}+4y^{\prime }=3t^3+t^2{e}^{-4t}+\text{sin}(2t)$

2. $y^{″}+4y=3t^3+t^2{e}^{-4t}+\text{sin}(2t)$

3. $y^{″}+4y^{\prime }+4y=3t^3+t^3{e}^{-2t}+\text{sin}(2t)$

4. $y^{″}+2y^{\prime }+2y=3t^3+t^2{e}^{-4t}+t\text{cos}(t)$

5. $y^{″}+2y^{\prime }+5y=3t^3+t^2{e}^{-4t}+e^{-t}\text{sin}(2t)$