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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.