Solve System of Differential Equations with Expert Help

Find the general solution to the equation ${\displaystyle x\left(\frac{dy}{dx}\right)+3(y+x^2)=\frac{\sin x}{x}}$

Use Laplace transform to solve the following initial-value problem
$y"+2y^{\prime }+y=0$
$y(0)=1,y^{\prime }(0)=1$
a) ${\displaystyle e^{-t}+t{e}^{-t}}$
b) ${\displaystyle e^t+2t{e}^t}$
c) ${\displaystyle e^{-t}+2t{e}^t}$
d) ${\displaystyle e^{-t}+2t{e}^{-t}}$
e) ${\displaystyle 2e^{-t}+2t{e}^{-t}}$
f) Non of the above

Solution of I.V.P for harmonic oscillator with driving force is given by Inverse Laplace transform
$y"+{\omega }^{2}y=\sin \gamma t,y(0)=0,y^{\prime }(0)=0$
1) $y(t)=L^{-1}(\frac{\gamma }{(s^2+{\omega }^2{)}^2})$
2) $y(t)=L^{-1}(\frac{\gamma }{s^2+{\omega }^2})$
3) $y(t)=L^{-1}(\frac{\gamma }{(s^2+{\gamma }^2{)}^2})$
4) $y(t)=L^{-1}(\frac{\gamma }{(s^2+{\gamma }^2)(s^2+{\omega }^2)})$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. 
$y"-4y\text{'}=-4t{e}^{2t},y_0=0,y{\text{'}}_0=1$

$y"+2y\text{'}+10y=-6e^{-t} \sin 3t,y_0=0,y{\text{'}}_0=1$

$y"-8y^{\prime }+41y=0$
$y(0)=0$
$y^{\prime }(0)=5$

Now, using Y for the Laplace transform of y(t), i.e., $Y=L\{y(t)\}$ find the equation that results from the differential equation's Laplace transformation.