Section 2.5. Find Solutions of Differential Equations using reduction of order

Objective:

1. Repeated root solutions with  $e^{at},t{e}^{at}$ for constant coefficients 2nd order differential equations

2. Use the reduction of order to find solution of non-constant coefficient 2nd order differential equations.

In section 2.4, we deal with distinct roots of the characteristic equation of the ODE, $a{y}^{″}+b{y}^{\prime }+cy=0$. In this section, we deal take care of the case when $b^2-4ac=0$ hence the characteristic equation has one repeated root. 

Case 3: $b^2-4ac=0,$then $\lambda =\frac{-b}{2a}=\alpha$ is a repeated root of $a{y}^{″}+b{y}^{\prime }+cy=0$. We know that $e^{\alpha t}$ is a solution of $a{y}^{″}+b{y}^{\prime }+cy=0$. We show $t{e}^{\alpha t}$ is also a solution of $a{y}^{″}+b{y}^{\prime }+cy=0.$ The Wronskian of $e^{\alpha t}$ and $t{e}^{\alpha t}$ is 

$W=\left[\begin{array}{cc}{e}^{\alpha t}& t{e}^{\alpha t}\\ \alpha e^{\alpha t}& e^{\alpha t}+\alpha t{e}^{\alpha t}\end{array}\right]=e^{2\alpha t}\ne 0.$

Hence $e^{\alpha t}$ and $t{e}^{\alpha t}$ is a fundament set of the solution of the ODE, $a{y}^{″}+b{y}^{\prime }+cy=0$.

Example 1: Find the general solution of the ODE, $y^{″}-2y^{\prime }+1y=0$.

Exercise 1: Find the general solution of the ODE, $y^{″}+4y^{\prime }+4y=0$.

Example 2: Find the solution of the IVP, $y^{″}+6y^{\prime }+9y=0$, $y(0)=2$, $y^{\prime }(0)=-1$.

Exercise 2: Find the solution of the IVP, $y^{″}-10y^{\prime }+25y=0$, $y(0)=1$, $y^{\prime }(0)=3$.

So far, we have find the general solutions of $a{y}^{″}+b{y}^{\prime }+cy=0$. How do we find solutions of general 2nd order differential equation, $y^{″}+p(t)y^{\prime }+q(t)y=0$? The $"t"$ in $t{e}^{\alpha t}$ is actually coming from a special method in the ODE, called reduction of order. Here, we use real example to show how to find the solution $t{e}^{\alpha t}$ in the above situation and then work on the case of $y^{″}+p(t)y^{\prime }+q(t)y=0$ when $p(t)$ and $q(t)$ are not constants. 

Example 3: Use reduction of order method to find the general solution of the ODE, $y^{″}-4y^{\prime }+4y=0$. 

Exercise 3: Use reduction of order to find the general solution of the ODE, $y^{″}+12y^{\prime }+36y=0$.

Example 4: Use reduction of order method to find the general solution of the ODE, $t^2{y}^{″}-5t{y}^{\prime }+8y=0$ with $y_1=t^2$.

Exercise 4: Use reduction of order to find the general solution of the ODE, $t^2{y}^{″}-4t{y}^{\prime }+4y=0$ with $y_1=t^4$.

Group Work:

1. Use reduction of order to find the general solution of the ODE, $t^2{y}^{″}+2t{y}^{\prime }-2y=0$ with $y_1=t$.

2. Use reduction of order to find the general solution of the ODE, $t^2{y}^{″}+3t{y}^{\prime }+y=0$ with $y_1=t^{-1}$.

3. Find the solution of the IVP, $y^{″}-4y^{\prime }+5y=0$, $y(0)=1$, $y^{\prime }(0)=0$.

4. Find the solution of the IVP, $y^{″}-4y^{\prime }=0$, $y^{\prime }(0)=0$, $y(0)=1$.

5. Find the solution of the IVP, $y^{″}-6y^{\prime }+9y=0$, $y(0)=1$, $y^{\prime }(0)=1$.