Write the given second order equation as its equivalent system of first order equations u''-8u'-4.5u=-3sin(3t), u(t)=6.5, u'(1)=8

If ${\displaystyle xy+6e^y=6e}$, find the value of y" at the point where x = 0.

A differential equation and a nontrivial solution f are given below. Find a second linearly independent solution using reduction of order. Assume that all constants of integration are zero.
$tx-(2t+1)x+2x=0,t>0,f(t)=3\cdot e^{2t}$

A particle moves along the curve ${\displaystyle x=2t^{2}y=t^2-4t}$ and z=3t-5 where t is the time.find the components of the velocity at t=1 in the direction i-3j+2k

Determine a unique solution of the separable differential equa- tion that satisfies the given initial condition.
${\displaystyle \frac{d^{2}y}{{dt}^2}=\frac{dy}{dt}\frac{dy}{dt}\left(0\right)=2,y\left(0\right)=3}$

Solve the equation:
${\displaystyle (a-x)dy+(a+y)dx=0}$

A dynamic system is represented by a second order linear differential equation.
$2\frac{d^{2}x}{d{t}^2}+5\frac{dx}{dt}-3x=0$
The initial conditions are given as:
when $t=0,x=4$ and $\frac{dx}{dt}=9$
Solve the differential equation and obtain the output of the system x(t) as afunction of t.

Solve the following second order Linear homogeneous /non-homogenous differential equation
${\displaystyle y{}^{″}-4y^{\prime }-12y=3e^{5x}}$