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 $y^{\prime }=ay$. 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 $y^{\prime }+p(t)y=g(t)$. First recall the product rule, $(u(t)f(t){)}^{\prime }=u(t)f^{\prime }(t)+u^{\prime }(t)f(t)$.

We can think $y^{\prime }=f^{\prime }(t)$ and $y=f(t)$. What is left from the left-hand side is $u(t)$ and $u^{\prime }(t)$. We assume $u(t)$ exists and we multiply $u(t)$ on both hand sides of the differential equation.

Definition:  Given a differential equation $y^{\prime }+p(t)y=g(t)$,  $u(t)=e^{\int p(t)dt}$ is the   integrating factor of the differential equation.

 Example 1: $y^{\prime }-2y=t^2$ then the integrating factor is $u(t)=e^{-2t}$. 

Exercise 1:    $y^{\prime }+3y=t$. Find the integrating factor.

Example 2: Solve the IVP: $y^{\prime }-2y=t$,  $y(0)=2$.

Exercise 2:    Solve the IVP: $y^{\prime }+3y=t$,  $y(0)=-1$.

  Example 3: $y^{\prime }-\frac{2}{t}y=t^2$,  $y(1)=3$. Solve the IVP.

Exercise 3: $y^{\prime }+\frac{3}{t}y=t^{-3}$ ,  $y(2)=-2$ solve the IVP.

Example 4: $t{y}^{\prime }+y=t\text{cos}(3t)$. Find general solution of the ODE. Describe the behavior of the solution as $t$ approaches infinity. 

Exercise 4: $t^2{y}^{\prime }-ty=t^3\text{sin}(2t)$ . Find general solution of the ODE. Describe the behavior of the solution as $t$ approaches infinity. 

 Group Work:

1. $y^{\prime }-y=1+\text{sin}(t)$ and $y(0)=y_0$. Find $y_0$ such that the solution of the ODE is finite as $t$ approaches infinity.

2. Solve the IVP.  $t{y}^{\prime }+(t+1)y=t$,  $y(\text{ln}(2))=1$. $t>0$

3. Solve the IVP.  $y^{\prime }+t^{-1}y=e^{-t}$,  $y(1)=1$. 

4. Solve the IVP.  $3y=t{y}^{\prime }+2t$,  $y(-1)=1$.