Section 1.4. Separable Differential Equations

Objective:

1. The definition of separable differential equation

2. Solve a separable differential equation

In this section, we learn to solve a new kind of differential equation, separable differential equation. Recall the population model, $y^{\prime }=ay$. This is a separable differential equation.

Definition: A first order differential equation, $y^{\prime }=\frac{dy}{dt}=f(y,t)$, is separable if it can be presented into

$$F(y)dy=G(t)dt$$

where $F(y)$ is a function of $y$ and $G(t)$ is a function of $t$.

Example 1: Find the general solution of the ODE: $y^{\prime }=\frac{y^2}{t}$.

Exercise 1: Find the general solution of the ODE: $y^{\prime }=\frac{x+2}{y^2}$ .

Example 2: Find the general solution of the ODE: $y^2{y}^{\prime }-\text{cos}(t)y=0$.

Exercise 2: Find the general solution of the ODE: $y{y}^{\prime }+e^t=0$.

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

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

Example 4: Solve the IVP: $e^y{y}^{\prime }-(x+\text{sin}(x))=0,y(0)=1$.

Exercise 4: Solve the IVP: $x^{-2}{y}^{\prime }-e^{x}y=0$, $y(0)=-2$.

Group work: Solve IVP

1. $y^{\prime }=\frac{t{y}^2}{\sqrt{1+t^2}},$$y(0)=2$.

2. $y^{\prime }=\frac{t^2}{3+2y}$, $y(1)=0$. 

3. $t{y}^{\prime }=\sqrt{1-y^2}$, $y(1)=1$.

4. $y^{\prime }=\frac{x^3}{3y^2+1}$, $y(1)=0$.

5. $y^{\prime }=ty+2$, $y(0)=3$.