Partial Fractions

Express $\displaystyle \frac{3x^{2}-3x+5}{x(x^{2}+5)}$ in partial fractions.

Express $\displaystyle \frac{3x^2 - x - 14}{(x + 3)(x - 1)^2}$ in partial fractions.

(a)Express $\displaystyle \frac{-2}{x(x+1)}$ in partial fractions.

(b)Use integration by parts to find $\displaystyle \int xe^{3x} dx.$

(c)Using your answers to (a) and (b), solve

$\displaystyle \frac{dy}{dx} - \frac{2y}{x(x+1)} = \frac{x^3 e^{3x}}{(x+1)^2}.$

Express $\displaystyle \frac{2x^2 - 18x + 4}{(x - 1)(x - 3)(x + 5)}$ in partial fractions.

(a)Express using partial fractions $\displaystyle \frac{x+3}{(x+7)(x+5)}.$

(b)Hence find the particular solution of the differential equation

$\displaystyle \frac{dy}{dx}+\frac{2}{x+3}y=\frac{1}{(x+7)(x+5)(x+3)},$ where $\displaystyle x\ge0,$

given that $\displaystyle y=0$ when $\displaystyle x=3.$