# Section 2: Separable Equations

### Instructions

• First, you should watch the concepts videos below explaining the topics in the section.
• Second, you should attempt to solve the exercises and then watch the videos explaining the exercises.
• Last, you should attempt to answer the self-assessment questions to determine how well you learned the material.
• When you have finished the material below, you can start on the next section or return to the main differential equations page.

### Concepts

• Definition of separable differential equations
• How to solve separable differential equations

### Exercises

Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Solve the differential equation $$y'=6y^2x.$$

$$y=\dfrac{1}{-3x^2+C}$$
or
$$y=0\quad$$ (an equilibrium solution)

2. Solve the initial value problem and determine the interval of validity$\sqrt{1+x^2}dy-xy^3dx=0,\quad y(0)=-1$

$$y=-\dfrac{1}{\sqrt{3-2\sqrt{1+x^2}}}\quad$$ I.V.: $$\left(-\dfrac{\sqrt{5}}{2}, \dfrac{\sqrt{5}}{2}\right)$$

### Self-Assessment Questions

Directions: The following questions are an assessment of your understanding of the material above. If you are not sure of the answers, you may need to rewatch the videos.
1. Is $$y'=x^2+y^2$$ separable? How about $$y'=x^2\cdot y^2$$?
2. Describe what an equilibrium solution of an ODE means.