Section 2.2 – Separable Equations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 

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Note: Problems 1 and 2 contain techniques from Sections 2.1 and 2.2.

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1. Find the general solution of the given differential equation. 
   1. \(y'+2ty=2te^{-t^2}\)
   2. \(2\sqrt{x}\,y'=\sqrt{1-y^2}\)
   3. \(ty'+y=3t\cos t,\qquad t>0\)​
      Answer

      1. \(y=(t^2+C)e^{-t^2}\)
      2. \(y=\sin(\sqrt{x}+C),\quad\) equilibrium solutions at \(y=\pm1\)
      3. \(y=\dfrac{3(t\sin(t)+\cos(t))+C}{t}\)

      Video

      
      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V9

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2. Find the solution to the initial value problem and the interval of validity in each case.
   1. \(2\sqrt{x}\dfrac{dy}{dx}=\cos^2 y,\qquad\) \(y(4)=\dfrac{\pi}{4}\)
   2. \(\dfrac{dy}{dt}+\dfrac{2y}{t}=\dfrac{\cos t}{t^2}\qquad\) \(y(1)=\dfrac{1}{2},\qquad\) \(t>0\)​
      Answer

      1. \(y=\arctan(\sqrt{x}-1), \quad\)\(I.V.=(0,\infty)\)
      2. \(y=\dfrac{\sin(t)+\dfrac{1}{2}-\sin(1)}{t^2}, \quad\) \(I.V. =(0,\infty)\)

      Video

      
      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V10

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