$2022-2023$ Differential Equations Final Exam

Solve the following differential equations by Undetermined Coefficient Method.

$\frac{d^{2}y}{d{x}^2}-4\frac{dy}{dx}+4y=16x^2{e}^{2x}$

Solve the linear system by Laplace Transformation.

$\frac{d^{2}y}{d{t}^2}+\frac{d^{2}x}{d{t}^2}=t^2$

$\frac{d^{2}x}{d{t}^2}-2\frac{d^{2}y}{d{t}^2}=4t$

$x(0)=8,y(0)=0$

$x^{\text{'}}(0)=0,y^{\text{'}}(0)=0$

Compute $f(t)=L^{-1}[F(s)]$

a$.$ by partial fractions

$F(s)=[\frac{s+1}{s^2+4s+9}]$

b$.$ by convolution

$F(s)=[\frac{4}{s^2+s-6}]$

A $12\mathrm{.}5m^3$ tank is being filled with water at a rate of $0\mathrm{.}05\frac{m^3}{s}.$ At a moment when the tank contains $1\mathrm{.}2m^3$ water, a bottom leak develops and gets progressively worse with time. The rate of leakage can be approximated as $0\mathrm{.}0025t(\frac{m^3}{s}),$ where $t(s)$ is the time from the moment leak begins.

a$.$ Find the expression for the volume of liquid water in terms of time using Laplace transforms.

b$.$ how long will it take for the compute emptying of tank$?$