Linear Ordinary Differential Equations:    General Case (how to solve)

A. Solve each linear differential equation.

  1. \(\displaystyle{x\frac{dy}{dx}- 4y=x^6e^x, \quad x>0}\)  solution

  2. \( \displaystyle{(x+1)\frac{dy}{dx}+(x+2)y = 2xe^{-x}, \quad x>-1}\)   solution

  3. \( \displaystyle{\frac{dr}{d\theta}+r\sec \theta = \cos\theta}, \quad 0 < \theta < \pi/2 \)   solution

  4. \( \displaystyle{x^2y'+xy=1}, \quad x > 0 \)  solution

  5. \( \displaystyle{\frac{dy}{dt}+\frac{1}{t} y = \cos t}, \quad t >0 \)   solution

  6. \( \displaystyle{x\frac{dy}{dx}- y = x^2\,\sin x}, \quad x>0 \)   solution

B. Solve the following initial value problems (IVP).

  1. \(\displaystyle{x\frac{dy}{dx}+2y=\frac{\sin x}{x}, \: y(\pi)=0, x > 0}\)   solution

  2. \( \displaystyle{\frac{dy}{dx}- (\sin x) y = 2 \sin x, \quad y(\pi/2)=1}\)   solution