Math 2565, Spring 2020

Show that \(y(t)=A\cos(2t)+B\sin(2t)\) is a solution to \(y''+4y=0\text{.}\)

Then find \(A\) and \(B\) such that \(y(0)=6\) and \(y'(0)=-3\text{.}\)

Classify each equation as separable, linear, both, or neither:

1. \(\frac{dy}{dx}=y-y^2\)

2. \(\left(\frac{dy}{dx}\right)^2=x^2e^y+2\)

3. \(\frac{dy}{dx}=\cos(x)+x^2y\)

4. \(\frac{dy}{dx}=xy-x\)

Examples:

1. \(\di \frac{dx}{dt}=(t^2-1)x\)

2. \(\di \frac{dx}{dt}=(x^2-1)t\)

3. \(\di \frac{dy}{dx} = xy+x+y+1\text{,}\) with \(y(0)=3\)

4. \(\di y' = \frac{\sin(x)}{\cos(y)}\) (implicit solution)

5. \(\di e^x yy' = e^{-y}+e^{-2x-y}\)

Examples:

1. \(y'+xy=x\)

2. \(y'+3x^2y = \sin(x)e^{-x^3}\text{,}\) with \(y(0)=1\)

3. \(y'+3x^2y = x^2\)

4. \(\di \frac{1}{x^2+1}y'+xy=3\text{,}\) with \(y(0)=0\)

5. \(xy'+4y=x^3-x\)