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

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. $$ty'+y=3t\cos t,\qquad t>0$$​

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

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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. $$\dfrac{dy}{dt}+\dfrac{2y}{t}=\dfrac{\cos t}{t^2}\qquad$$ $$y(1)=\dfrac{1}{2},\qquad$$ $$t>0$$​

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

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