Find a differential equation whose general solution is y=c1e−t/2+c2e−2t.

Find the scalars c1, c2, and c3 such that

$$c_1(1,-1,0) + c_2(3,2,1) + c_3(0,3,1) = (1,2,1)$$

Find a differential equation whose general solution is

$$\begin{gather*} y = c_{1} e^{-3t} \cos 4t + c_{2} e^{-3t} \sin 4t. \end{gather*}$$

Solve the initial value problem.

$$\begin{gather*} y'' + 16y = 0, \,\,\, y(0) = 0, \,\,\, y'(0) = 1, \end{gather*}$$

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

Determine the form of a particular solution for the differential equation. Do not solve:

$$y'' - 4y' + 5y = e^{5t} + t \sin3t - \cos3t$$

letφ0(t)=0 and use the method of successive approximations to approximate the solution of the given initial value problem.(a) Calculateφ1(t),...,φ4(t), or (if necessary) Taylor approximations to these iterates. Keep terms up to order six.(b) Plot the functions you found in part (a) and observe whether they appear to be converging. y'=(3t2+4t+2)/2(y−1),y(0)=0

find the fundamental set of solutions specified by Theorem for the given differential equation and initial point. y"+4y'+3y=0,t0=1

find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as t increases. 2y"+y'−4y=0,y(0)=0,y'(0)=1

(a) Find the solution of the given initial value problem in explicit form.(b) Plot the graph of the solution.(c) Determine (at least approximately) the interval in which the solution is defined.y'=2x/(1+2y),y(2)=0

find the solution of the given initial value problem. Sketch the graph of the solution and describe its behavior as t increases.y”+4y'+3y=0,y(0)=2,y'(0)=−1

find the general solution of the given differential equation. 9y"+9y'−4y=0

find the general solution of the given differential equation. 4y"−9y=0

y''+ay'+by=0 for the given basis. cos 2πx, sin 2πx.

find the solution of the given initial value problem. y' − y = 2te2 t, y (0) = 1