Solve System of Differential Equations with Expert Help

${\displaystyle y^{\prime }=6\frac{x^2}{2y+\cos y}}$

${\displaystyle \left(b\right)y+y\prime +3.25=0}$

Solve the linear equations by considering y as a function of x, that is, $y=y(x)$.
$y^{\prime }+2y=4$

Solve the following IVP using Laplace Transform
$y\prime \prime +3y\prime +2y=e^{-t},y(0)=0y\prime (0)=0$

How to solve for third order differential equation of $y"-7y\text{'}+6y=2\sin \left(t\right)$ using Method of Laplace Transform when $y\left(0\right)=0,y\text{'}\left(0\right)=0,y"\left(0\right)=0$?
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Solve differential equation $y^{\prime }(t)=-3y+9$, $y(0)=4$

Solve differential equation $\frac{dy}{dx}+(\frac{a}{x})y=40x$, for $x>0$ and $y(1)=a$

Solve the differential equation $y^{\prime }-\lambda y=1-\lambda t$,  $y(0)=0$

use the Laplace transform to solve the initial value problem.
$y"-3y^{\prime }+2y=\{\begin{array}{ll}0& 0\le t<1\\ 1& 1\le t<2\\ -1& t\ge 2\end{array}$
$y(0)=-3$
$y^{\prime }(0)=1$

Solve differential equation $(1+y^2+x{y}^2)dx+(x^{2}y+y+2xy)dy=0$

Solve the correct answer linear equations by considering y as a function of x, that is, ${\displaystyle y=y\left(x\right).\frac{dy}{dx}+y=\cos x}$

Solve differential equation $\frac{{\cos }^{2}y}{4x+2}dy=\frac{(\cos y+\sin y{)}^2}{\sqrt{x^2+x+3}}dx$