Section 2.2 – Separable Equations
Note: Problems 1 and 2 contain techniques from Sections 2.1 and 2.2.
Exercises
Directions: You should try to solve each problem first, and then click "Reveal Answer" to check your answer. You can click "Watch Video" if you need help with a problem.
1. Find the general solution of the given differential equation.
- \(y'+2ty=2te^{-t^2}\)
- \(2\sqrt{x}\,y'=\sqrt{1-y^2}\)
- \(ty'+y=3t\cos t,\qquad t>0\)
- \(y=(t^2+C)e^{-t^2}\)
- \(y=\sin(\sqrt{x}+C),\quad\) equilibrium solutions at \(y=\pm1\)
- \(y=\dfrac{3(t\sin(t)+\cos(t))+C}{t}\)
2. Find the solution to the initial value problem and the interval of validity in each case.
- \(2\sqrt{x}\dfrac{dy}{dx}=\cos^2 y,\qquad\) \(y(4)=\dfrac{\pi}{4}\)
- \(\dfrac{dy}{dt}+\dfrac{2y}{t}=\dfrac{\cos t}{t^2}\qquad\) \(y(1)=\dfrac{1}{2},\qquad\) \(t>0\)
- \(y=\arctan(\sqrt{x}-1), \quad\)\(I.V.=(0,\infty)\)
- \(y=\dfrac{\sin(t)+\dfrac{1}{2}-\sin(1)}{t^2}, \quad\) \(I.V. =(0,\infty)\)