Solve System of Differential Equations with Expert Help

$x^2+y^2=25$
What is dydx dx/dy?

Solve the following non-homogeneous second order linear differential equation.
$y^{″}+p{y}^{\prime }-qy=-rt$
if $p=4,q=8,r=7$.

Transform:
${\displaystyle f\left(t\right)={\cos }^{2}at}$

In an integro-differential equation, the unknown dependent variable y appears within an integral, and its derivative $\frac{dy}{dt}$ also appears. Consider the following initial value problem, defined for t > 0:
$\frac{dy}{dt}+4{\int }_0^{t}y(t-w)e^{-4w}dw=3,y\left(0\right)=0$
a) Use convolution and Laplace transforms to find the Laplace transform of the solution.
$Y\left(s\right)=L\left\{y\left(t\right)\right)\}-?$
b) Obtain the solution y(t).
y(t) - ? 

Q. 2# $(x+1)\frac{dy}{dx}=x(y^2+1)$

Solve the differential equation$2y"+20y^{\prime }+51y=0,y(0)=2$
$y^{\prime }(0)=0$

Determine the first derivative ${\displaystyle \left(\frac{dy}{dx}\right)ofy=\tan \left(3x^2\right)}$

Use the Laplace transform to solve the following initial value problem:
$2y"+4y^{\prime }+17y=3\cos (2t)$
$y(0)=y^{\prime }(0)=0$
a)take Laplace transform of both sides of the given differntial equation to create corresponding algebraic equation and then solve for $L\{y(t)\}$b) Express the solution $y(t)$ in terms of a convolution integral

Use the Laplace transform table and the linearity of the Laplace transform to determine the following transform.
$L\{e^{3t}\sin (4t)-t^4+e^t\}$

Find the inverse Laplace transform
${\displaystyle \frac{4s^3-s^2+5s+2}{s^4}}$

Find the inverse Laplace transform of $F(s)=\frac{(s+4)}{(s^2+9)}$
a)$\cos (t)+\frac{4}{3}\sin (t)$
b)non of the above
c) $\cos (3t)+\sin (3t)$
d) $\cos (3t)+\frac{4}{3}\sin (3t)$
e)$\cos (3t)+\frac{2}{3}\sin (3t)$
f) $\cos (t)+4\sin (t)$

Use Laplace Transform to solve the given equation:
$y^{″}+6y^{\prime }+5y=t\cdot U(t-2)$
$y(0)=1$
$y^{\prime }(0)=0$

Solve the question with laplace transformation
$y^{″}+2y^{\prime }+y=4$
$y(0)=3$
$y^{\prime }(0)=0$

Solve the following equation with Laplace Transform Method (Inverse Laplace the equation to find the solution)
$y"-3y^{\prime }-4y=3e^{2x}$
$y(0)=1,y^{\prime }(0)=0$

Solve the differential equation:
$y^{″}+6y^{\prime }+12y=0$

Solve differential equation$y^{\prime }(t)=-3y+9$, y(0)=4

Use the Laplace Transform solve the IVP
\(\begin{cases}x+y=t\

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.
$tx-(2t+1)x+2x=0,t>0,f(t)=3\cdot e^{2t}$

Derive the Laplace transform of the following function
$\sin (h(at))$

Solve differential equation $((3y^2-t^2)/y^5)dy/dx+t/(2y^4)=0$, y(1)=1