# Toronto Math Forum

## MAT244--2018F => MAT244--Tests => Term Test 2 => Topic started by: Victor Ivrii on November 20, 2018, 05:51:00 AM

Title: TT2A-P2
Post by: Victor Ivrii on November 20, 2018, 05:51:00 AM
Consider equation

y'''-2y''  -y'+2y= 8e^{t}.
\label{2-1}

(a)
Write a differential equation for the Wronskian of $y_1,y_2,y_3$, which are solutions for homogeneous equation and solve it.

(b) Find fundamental system $\{y_1,y_2,y_3\}$ of solutions for homogeneous equation, and find their Wronskian. Compare with (a).

(c) Find the general solution of (\ref{2-1}).
Title: Re: TT2A-P2
Post by: Zhihao Zuo on November 20, 2018, 08:25:28 AM
This is my solution, thanks!
Title: Re: TT2A-P2
Post by: Samarth Agarwal on November 20, 2018, 09:56:01 AM
Here we can use Abel's Theorem to determine the value of the Wronskian
$$\mbox{Therefore, } W = c e^{2t}, since p(t) = -2$$

for part b, we must find the characteristic equation
$$r^3 - 2r^2 - r + 2 = 0$$
$$(r-1)(r+1)(r-2) = 0$$
$$\mbox{Therefore, } r = 1, -1, 2$$
$$\mbox{Therefore, } y_1(t) = e^{t} y_2(t) = e^{-t} y_3(t) = e^{2t}$$
$$\mbox{Therefore, the Wronskian} W(y_1, y_2, y_3)(t) = \begin{bmatrix} e^{t}&e^{-t}&e^{2t}\\ e^{t}&-e^{-t}&2e^{2t}\\e^{t}&e^{-t}&4e^{2t}\\ \end{bmatrix}$$
$$det(\begin{bmatrix} e^{t}&e^{-t}&e^{2t}\\ e^{t}&-e^{-t}&2e^{2t}\\e^{t}&e^{-t}&4e^{2t}\\ \end{bmatrix}) = -6 e^{2t}$$
This is similiar to the solution found in part a, except that c = -6

for part c,
$$\mbox{Assume} y(t) = Ate^{t}$$
$$\mbox{Therefore, } y'(t) = Ae^{t}(1+t)$$
$$\mbox{Therefore, } y''(t) = Ae^{t}(2+t)$$
$$\mbox{Therefore, } y'''(t) = Ae^{t}(3+t)$$
Plugging this in the original equation
$$Ae^{t}(t + 3 - 4- 2t - 1 - t - 2t) = 8Ae^{t}$$
$$\mbox{Therefore, A = -4}$$
$$\mbox{Therefore, the general equation is } y(t) = c_1e^{t} + c_2e^{-t} + c_3e^{2t} -4te^{t}$$