Solve the equation.delta(t-t_0) is the Dirac-Delta function. y"+4y'+5y=delta(t-2pi)+delta(t-4pi) , y(0)=0 , y'(0)=0

Solve the following linear differential equation ${\displaystyle y^6-64y=0}$

Find the inverse Laplace transform of the following:
$\left(\frac{7}{2}\right)\log \left(\frac{s-8}{s+8}\right)$

Verify that each given function is a solution of the differential equation.
${\displaystyle y2y^{\prime }-3y=0;y_1\left(t\right)=e^{-3t},y_2\left(t\right)=e^t}$

Find the solution of the following Differential Equations ${\displaystyle y-y^{\prime }-6y=0,y\left(0\right)=6,y^{\prime }\left(0\right)=13}$.

Classify the following differential equations as separable, homogeneous, parallel line, or exact. Explain briefly your answers. Then, solve each equation according to their classification. ${\displaystyle 2x{\sin }^{2}ydx-(x^2-9)\cos ydy=0}$

Find the inverse Laplace transform $f(t)=L^{-1}\{F(s)\}$ of the function ${\displaystyle F\left(s\right)=\frac{8e^{-4s}}{s^2+64}}$
Use h(t-c) for the Heaviside function h_c(t) if necessary.
$f(t)=L^{-1}\left\{\frac{8e^{-4s}}{s^2+64}\right\}=$

Implicit differentiation question
Given: ${\displaystyle \frac{y}{x-y}=x^2+1}$