Problem $1$

Find value of $m$ that function $y=x^m$ is a solution of the differential equation:

$x^2{y}^{\text{'}\text{'}}-7x{y}^{\text{'}}15y=0$

$(15$ points $)$

Problem $2$

A general solution of the differential equation: $xx=0$ is represented by the two$-$parameter family of solutions: $x=c_{1}cos(t)c_{2}sin(t).$ Find the solution of the initial value problem for $x(0)=-1,$ and $x^{\text{'}}(0)=8.$

$(25$ points$)$

Problem $3$

Verify that y is a solution of the ODE, C being a constant.

a$.y^{\text{'}}4y=1\mathrm{.}4,y=C{e}^{-4x}$

b$.y^{\text{'}}=y{e}^x,y=(xC)e^x$

c$.y^{\text{'}}tan(x)=2y-8,$ if $y=Csi{n}^2(x)4$

$(30$ points$)$