Section 2.7. Solutions of non-homogeneous differential equations

Objective:

1. Definition of the complementary solution and a particular solution of a differential equation

2. Find the solution of a non-homogeneous differential equation

Recall for Section 2.1, this whole chapter is trying to solve the 2nd order differential equations, $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$. When $g(t)$ is zero, this type of differential equation is called homogeneous. When $g(t)$ is not zero, this type of differential equation is called non-homogenous. 

Definition: Given a differential equation, $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$, the general solution of $y^{″}+p(t)y^{\prime }+q(t)y=0$ is called the complementary solution of the ODE. A particular $y_p$ such that $y_p^{″}+p(t)y_p^{\prime }+q(t)y_p=g(t)$ is called a particular solution of the ODE.

Example 1: Find the complementary solution of the ODE and a particular solution of the ODE. $y^{″}+y^{\prime }-2y=e^{2t}$.

Exercise 1: Find the complementary solution of the ODE and a particular solution of the ODE. $y^{″}-y^{\prime }-6y=e^{4t}$.

Theorem: If $Y_1$ and $Y_2$ are two solutions of $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$, then $Y_1-Y_2$ is a solution of $y^{″}+p(t)y^{\prime }+q(t)y=0$. In particular, if $y_1$ and $y_2$ form a fundamental set of solutions of $y^{″}+p(t)y^{\prime }+q(t)y=0$, then there are $c_1$ and $c_2$ such that $Y_1-Y_2=c_1{y}_1+c_2{y}_2$.

Proof: 

Theorem: The general solution of $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$ is $y=y_c+y_p$ where $y_c$ is the general solution of $y^{″}+p(t)y^{\prime }+q(t)y=0$ and $y_p$ is a particular solution of $y^{″}+p(t)y^{\prime }+q(t)y=g(t)$. 

Proof:

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

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

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

Exercise 3: Find a general solution of $y^{″}+6y^{\prime }+9y=4t^2-1$.

Example 4: Find a general solution of $y^{″}+4y^{\prime }=\text{cos}(2t)$. 

Exercise 4: Find a general solution of $y^{″}+9y=\text{sin}(5t)$.

Group Work

1. Find the solution of the IVP: $y^{″}-2y^{\prime }-3y=e^{2t}+\text{cos}(t)$, $y(0)=1,$ and $y^{\prime }(0)=-1$.

2. Find a general solution of $y^{″}+y^{\prime }+4y=\text{cos}(3t)+t$. 

3. Find a general solution of $y^{″}-4y^{\prime }-5y=t{e}^{-3t}$.

4. Find a general solution of $y^{″}-4y^{\prime }+4y=t\text{sin}(t)$.