Question

Find the general solution of the following differential equation x dy / dt - 4y = x^5 e^x 

Answer 1

$$x\dfrac{dy}{dx}-4y=x^5e^x$$

$$y'-\dfrac{4}{x}y=x^4 e^x$$

Find the integrating factor

$$\mu(x)=\dfrac{1}{x^4}$$

$$(\dfrac{1}{x^4}y)'=e^x$$

$$\dfrac{1}{x^4}y=e^x+c_1$$

$$y=x^4e^x+c_1x^4$$