Differential Equations

Find the particular solution of the differential equation

$\displaystyle \frac{d^2y}{dx^2} - 6\frac{dy}{dx} + 9y = 8\sin x + 19\cos x$

given that $\displaystyle y = 7$ and $\displaystyle \frac{dy}{dx} = \frac{1}{2}$ when $\displaystyle x = 0.$

An electronic device contains a timer circuit that switches off when the voltage, $\displaystyle V$, reaches a set value.
The rate of change of the voltage is given by

$\displaystyle \frac{dV}{dt} = k(12 - V)$,

where $\displaystyle k$ is a constant, $\displaystyle t$ is the time in seconds, and $\displaystyle 0 \le V < 12.$
Given that $\displaystyle V = 2$ when $\displaystyle t = 0$, express $\displaystyle V$ in terms of $\displaystyle k$ and $\displaystyle t.$

Find the particular solution of the differential equation

$\displaystyle \frac{d^2y}{dx^2} + \frac{dy}{dx} - 6y = 35e^{2x}$

given $\displaystyle y = 5$ and $\displaystyle \frac{dy}{dx} = 12$ when $\displaystyle x = 0.$

(a)Differentiate $\displaystyle x \ln x - x$ with respect to $\displaystyle x.$

(b)Hence find the general solution of the differential equation $\displaystyle \frac{dy}{dx} + y \ln x = x^{-x}.$

Find the particular solution of the differential equation
$\displaystyle \frac{d^2y}{dx^2} - 4\frac{dy}{dx} - 5y = 10x^2 + 11x - 23$
given that $\displaystyle y = 2$, $\displaystyle \frac{dy}{dx} = 14$ when $\displaystyle x = 0.$

Points scored in the long jump element of the decathlon can be calculated using a solution of the differential equation

$\displaystyle \frac{dP}{dm} = \frac{1.4P}{m-220}$, $\displaystyle m > 220$

where $\displaystyle m$ is the distance jumped in centimetres and $\displaystyle P$ the points scored.

Given that a jump of 807 centimetres scores 1079 points, find an expression for $\displaystyle P$ in terms of $\displaystyle m.$

Find the general solution of the differential equation

$\displaystyle \frac{dy}{dx} - \frac{2}{x}y = x^2 \sec^2 3x.$

Give your answer in the form $\displaystyle y = f(x).$

Find the particular solution of the differential equation

$\displaystyle 2\frac{d^{2}y}{dx^{2}}-3\frac{dy}{dx}+y=0$

given that $\displaystyle y=2$ and $\displaystyle \frac{dy}{dx}=-1$ when $\displaystyle x=0.$