Question Type 4: Solving with the integrating factor (linear ODEs) Exercises

Solve the differential equation $y' + \dfrac{1}{x} y = \dfrac{\sin x}{x}$ for $x > 0$, given that $y(\pi)=0$.

Solve the differential equation $y' = \frac{dy}{dx} = \cos x + y \tan x$ for $y(x)$.

Solve $y' + \dfrac{3}{x} y = 4x^2$ for the general solution $y(x)$ (assume $x>0$).

Solve the ODE $y' + (1 - \tan x)\,y = \sec x$ for the general solution $y(x)$.

Solve $x\,y' + y = x^2$ with $y(1)=1$. [6 marks]

Solve $y' + 2y = e^{-x}$ with the initial condition $y(0)=0$.

Solve $y' - y = e^{2x}$ with $y(0)=0$.

Solve $y' + \dfrac{2}{x} y = 3x$ with $y(1)=2$.

Solve $y' + 2xy = x$ for the general solution $y(x)$.

Solve $y' + \dfrac{1}{1+x}y = \dfrac{2}{1+x}$ subject to $y(0)=0$.

Solve the differential equation $y' = \cos(x) + y\tan(x),$ given the boundary condition $y(0)=1$.

Solve $y' - \dfrac{2}{x} y = x^2$ subject to $y(1)=0$.