Solve dfrac{d^{2}y}{dt^{2}} - 8dfrac{dy}{dt} + 15y=9te^{3t} with y(0)=5, y'(0)=10

Find the general solutions of the differential equations ${\displaystyle 6y^4+11y4y=0}$

Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. ${\displaystyle (2x-5y+1)dx-(5y-2x)dy=0}$

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

The Laplace transform of $tu(t-2)=e^{-2s}\left[\frac{s+2s^2}{s^3}\right]$
True or False?

solve the IVP $y"+2y^{\prime }+17y=0.y(0)=1,y^{\prime }(0)=-1$
Thanks in advance.
 

Consider the linear first order non-homogeneous partial differential equation
$U_x+y{U}_y-y=y{e}^{-x}$
By using the method of characteristics show that its general solution is given by
$u(x,y)=-y{e}^{-x}+e^{x}g(y{e}^{-x})$
where g is any differentiable function of its argument
$$\begin{gathered} A=1,B=y,d=-1,f=y{e}^{-x} \\ {\displaystyle \frac{dy}{dx}}={\displaystyle \frac{b}{a}}=y \\ dy=ydx \\ \int {\displaystyle \frac{1}{y}}dy=\int dx \\ \ln y=x+c \\ \eta (x,y)=c \\ \text{let }\xi =x \end{gathered}$$
Then
$\eta =\ln y-x$
$\xi =x$
those are my characteristics curves,
when I put it into $[a{\xi }_x+b{\xi }_y]{\omega }_{\xi }+d\omega =F$
I get the equation ${\omega }_{\xi }-\omega =e^{\eta }$
Is this all correct?

Let $l(x)$ be the linear approximation of $f(x)=x^{2/5}$ at $a=32$. Approximation?