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

Objective:

1. Repeated root solutions with  [latex]e^{at},te^{at}[/latex] 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, [latex]ay''+by'+cy=0[/latex]. In this section, we deal take care of the case when [latex]b^{2}-4ac=0[/latex] hence the characteristic equation has one repeated root. 

Case 3: [latex]b^{2}-4ac=0,[/latex]then [latex]\lambda=\frac{-b}{2a}=\alpha[/latex] is a repeated root of [latex]ay''+by'+cy=0[/latex]. We know that [latex]e^{\alpha t}[/latex] is a solution of [latex]ay''+by'+cy=0[/latex]. We show [latex]te^{\alpha t}[/latex] is also a solution of [latex]ay''+by'+cy=0.[/latex] The Wronskian of [latex]e^{\alpha t}[/latex] and [latex]te^{\alpha t}[/latex] is 

[latex]W=\left[\begin{array}{cc} e^{\alpha t} & te^{\alpha t}\\ \alpha e^{\alpha t} & e^{\alpha t}+\alpha te^{\alpha t} \end{array}\right]=e^{2\alpha t}\neq0.[/latex]

Hence [latex]e^{\alpha t}[/latex] and [latex]te^{\alpha t}[/latex] is a fundament set of the solution of the ODE, [latex]ay''+by'+cy=0[/latex].

Example 1: Find the general solution of the ODE, [latex]y''-2y'+1y=0[/latex].

Exercise 1: Find the general solution of the ODE, [latex]y''+4y'+4y=0[/latex].

Example 2: Find the solution of the IVP, [latex]y''+6y'+9y=0[/latex], [latex]y(0)=2[/latex], [latex]y'(0)=-1[/latex].

Exercise 2: Find the solution of the IVP, [latex]y''-10y'+25y=0[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=3[/latex].

So far, we have find the general solutions of [latex]ay''+by'+cy=0[/latex]. How do we find solutions of general 2nd order differential equation, [latex]y''+p(t)y'+q(t)y=0[/latex]? The [latex]"t"[/latex] in [latex]te^{\alpha t}[/latex] 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 [latex]te^{\alpha t}[/latex] in the above situation and then work on the case of [latex]y''+p(t)y'+q(t)y=0[/latex] when [latex]p(t)[/latex] and [latex]q(t)[/latex] are not constants. 

Example 3: Use reduction of order method to find the general solution of the ODE, [latex]y''-4y'+4y=0[/latex]. 

Exercise 3: Use reduction of order to find the general solution of the ODE, [latex]y''+12y'+36y=0[/latex].

Example 4: Use reduction of order method to find the general solution of the ODE, [latex]t^{2}y''-5ty'+8y=0[/latex] with [latex]y_{1}=t^{2}[/latex].

Exercise 4: Use reduction of order to find the general solution of the ODE, [latex]t^{2}y''-4ty'+4y=0[/latex] with [latex]y_{1}=t^{4}[/latex].

Group Work:

1. Use reduction of order to find the general solution of the ODE, [latex]t^{2}y''+2ty'-2y=0[/latex] with [latex]y_{1}=t[/latex].

2. Use reduction of order to find the general solution of the ODE, [latex]t^{2}y''+3ty'+y=0[/latex] with [latex]y_{1}=t^{-1}[/latex].

3. Find the solution of the IVP, [latex]y''-4y'+5y=0[/latex], [latex]y(0)=1[/latex], [latex]y'(0)=0[/latex].

4. Find the solution of the IVP, [latex]y''-4y'=0[/latex], [latex]y’(0)=0[/latex], [latex]y(0)=1[/latex].

5. Find the solution of the IVP, [latex]y''-6y'+9y=0[/latex], [latex]y(0)=1[/latex], [latex]y’(0)=1[/latex].