1Obtain the General Solution of the differential equations:

(i) $\displaystyle x\frac{dy}{dx} + y = xe^{3x}$

(ii) $\displaystyle \frac{d^2y}{dx^2}=5\frac{dy}{dx}+6y=10e^{-2x}$

2(a) Find the general solution of the differential equation

$\displaystyle x\frac{dy}{dx}+2y=({1+\frac{2}{x}})e^x$

(b) Find the solution of

$\displaystyle \frac{d^2y}{dx^2}-4\frac{dy}{dx}+5y=2e^{3x}$

given that at x=0,y=2 and $\displaystyle \frac{dy}{dx}=4$

3(a) Find the following two integrals:

$\displaystyle \int\frac{x}{1-x^2}dx$ and $\displaystyle \int\frac{x}{1-x^2}dx$

Find y in terms of x, givene that

$\displaystyle \frac{dx}{dx}-\frac{x}{1-x^2}y=\frac{x}{(1-x^2)^{\frac{5}{2}}}$

and that y =1 when x = 0.

(b) Solve the differential equation

$\displaystyle \frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0$

given that y=1 and $\displaystyle \frac{dy}{dx}=3$ when x=0.