find the general solution of the given differential equation. y” − 2y' − 3y = 3e2t

find the general solution of the given differential equation . y” − y' − 2y = −2t + 4t2

determine the general solution of the given differential equation. y'''−y'=2 sint

verify that each given function is a solution of the given partial differential equation. α2uxx=ut;u=(π/t)1/2e−x2/4α2t,t>0

determine the general solution of the given differential equation. y(4)+y'''=sin 2t

use the method of variation of parameters to determine the general solution of the given differential equation. y(4)+2y''+y=sint

Find the general solution of the given differential equation. Give the largest interval over which the general solution is defined. Determine whether there are any transient terms in the general solution.

$$\frac { d y } { d x } = 5 y$$

find the general solution of the given differential equation. y” + 2y' + 5y = 3 sin 2t

In each of Problems 1 through 6: (a) Express the general solution of the given system of equations in terms of real-valued functions. (b) Also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as t → ∞.

determine the general solution of the given differential equation.y”’−y”−y'+y=2e−t+3

Solve the initial value problem.

$$\begin{gather*} y'' + 4y' + 4y = 0, \,\,\, y(-1) = 2, \,\,\, y'(-1) = 1, \end{gather*}$$

sketch the graph of its solution and describe its behavior for increasing ${t}$.

find the general solution of the given differential equation. y” +2y' = 3 + 4 sin 2t

Use the method of reduction of order to find a second solution of the given differential equation.

$$t^2y” − 4ty' + 6y =0, t > 0; \ y_1(t) = t^2$$

find the solution of the given initial value problem. y” −2y' + y= tet+4, y(0) = 1, y'(0) = 1