Section 1.3. The Integrating Factors
Objective:
1. The definition of integrating factor
2. Using integrating factor to solve the first order DE
Recall that the population model is [latex]y'=ay[/latex]. In this section we are going to solve this type of differential equation using integrating factors. In fact, we are going to solve the first order linear differential equation [latex]y'+p(t)y=g(t)[/latex]. First recall the product rule, [latex](u(t)f(t))'=u(t)f'(t)+u'(t)f(t)[/latex].
We can think [latex]y'=f'(t)[/latex] and [latex]y=f(t)[/latex]. What is left from the left-hand side is [latex]u(t)[/latex] and [latex]u'(t)[/latex]. We assume [latex]u(t)[/latex] exists and we multiply [latex]u(t)[/latex] on both hand sides of the differential equation.
Definition: Given a differential equation [latex]y'+p(t)y=g(t)[/latex], [latex]u(t)=e^{\int p(t)dt}[/latex] is the integrating factor of the differential equation.
Example 1: [latex]y'-2y=t^{2}[/latex] then the integrating factor is [latex]u(t)=e^{-2t}[/latex].
Exercise 1: [latex]y'+3y=t[/latex]. Find the integrating factor.
Example 2: Solve the IVP: [latex]y'-2y=t[/latex], [latex]y(0)=2[/latex].
Exercise 2: Solve the IVP: [latex]y'+3y=t[/latex], [latex]y(0)=-1[/latex].
Example 3: [latex]y'-\frac{2}{t}y=t^{2}[/latex], [latex]y(1)=3[/latex]. Solve the IVP.
Exercise 3: [latex]y'+\frac{3}{t}y=t^{-3}[/latex] , [latex]y(2)=-2[/latex] solve the IVP.
Example 4: [latex]ty'+y=t\text{cos}(3t)[/latex]. Find general solution of the ODE. Describe the behavior of the solution as [latex]t[/latex] approaches infinity.
Exercise 4: [latex]t^{2}y'-ty=t^{3}\text{sin}(2t)[/latex] . Find general solution of the ODE. Describe the behavior of the solution as [latex]t[/latex] approaches infinity.
Group Work:
1. [latex]y'-y=1+\text{sin}(t)[/latex] and [latex]y(0)=y_{0}[/latex]. Find [latex]y_{0}[/latex] such that the solution of the ODE is finite as [latex]t[/latex] approaches infinity.
2. Solve the IVP. [latex]ty'+(t+1)y=t[/latex], [latex]y(\text{ln}(2))=1[/latex]. [latex]t>0[/latex]
3. Solve the IVP. [latex]y'+t^{-1}y=e^{-t}[/latex], [latex]y(1)=1[/latex].
4. Solve the IVP. [latex]3y=ty'+2t[/latex], [latex]y(-1)=1[/latex].