Solve the initial-value problem

$$y''+5y'+4y=16x+20e^x,y(0)=0,y'(0)=3$$

Show that when light reflects from two mirrors that meet each other at a right angle, the outgoing ray is parallel to the incoming ray.

Use the Laplace transform to solve the given initial-value problem.

Use the Laplace transform to solve the given initial-value problem.

Use the Laplace transform to solve the given initial value problem

$$\begin{gather*} y^{(4)} - 4y''' + 6y'' - 4y' + y = 0; \\ y(0) = 0, \,\,\, y'(0) = 1, \,\,\, y''(0) = 0, y'''(0) = 1. \end{gather*}$$

Solve the initial-value problem.

$$y’’-5y’+4y=0, y(0)=0, y’(0)=1$$

Use the Laplace transform to solve the given equation.

Use the Laplace transform to solve the given equation.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions

$$y^{^{\prime \prime }}-4y^{^{\prime }}+4y=4,~~~~~y_{0}=0,~y_{0}^{^{\prime }}=-2.$$

Transform the initial value problem

$$\begin{gather*} y''- 5y' - 6y = 0, \,\,\,\,\,\, y(0) = 1, \,\,\, y'(0) = 0, \end{gather*}$$

into an algebraic equation for ${Y = L[y]}$ in the s-domain. Then find the Laplace transform of the solution of the initial value problem.

Solve the IVP $y^{\prime \prime}+3 y^{\prime}+2 y=6 ; \quad y(0)=0, \quad y^{\prime}(0)=2$ using Laplace transforms.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions

$$y^{^{\prime \prime }}+5y^{^{\prime }}+6y=12,~~~~~y_{0}=2,~y_{0}^{^{\prime }}=0.$$

By using Laplace transforms, solve the following differential equations subject to the given initial conditions

$$y^{^{\prime \prime }}+4y^{^{\prime }}+4y=e^{-2t},~~~~~y_{0}=0,~y_{0}^{^{\prime }}=4.$$

Simplify. Express your answer in scientific notation. Round to the nearest hundredth. There are approximately

$$10(4^9)^2$$

stars in the Milky Way Galaxy.