# DE exam review questions help

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• Jul 27th 2008, 08:29 AM
DE novice
DE exam review questions help
Help ! Have 2 problems on tomorrow's exam similar to these 2 and I haven't been able to get anywhere with them in 2 days ! Please show me the steps so I don't bomb ! Thanks !

1)
Find the solution to the system:

x'-2x-y'-y=6*exp^(3t)
2x'-3x+y'-3y=6*exp^(3t)

satisfying the condition x(0)=3 and y(0)=0

2)
solve
y'+4y=4t^2-4t+10
• Jul 27th 2008, 09:20 AM
Matt Westwood
No. 2 is straightforward enough, it's in the form y' + py = q where p and q are functions of t only (p in fact being a constant here).

So it's an integrating factor job - multiply through by e^(integral of p with respect to t).

Then the LHS is d/dt (ye^(that integral)) and then it's just a matter of integrating both sides w.r.t. t and the job's done.
• Jul 28th 2008, 12:58 PM
TwistedOne151
Problem 1
On number one,
First, you'll want to solve the equations for x' and y'. So adding the two equations together, we get:
$3x'-5x-4y=12e^{3t}$
$x'-\frac{5}{3}x-\frac{4}{3}y=4e^{3t}$
$x'=\frac{5}{3}x+\frac{4}{3}y+4e^{3t}$
And if we subtract twice the first equation from the second, we find:
$3y'+x-y=-6e^{3t}$
$y'+\frac{1}{3}x-\frac{1}{3}y=-2e^{3t}$
$y'=-\frac{1}{3}x+\frac{1}{3}y-2e^{3t}$

If we define the vectors $\mathbf{x}(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatri x}$, $\mathbf{b}(t)=\begin{pmatrix}4e^{3t}\\-2e^{3t}\end{pmatrix}$, and the matrix $\mathbf{A}=\begin{pmatrix}\frac{5}{3}&\frac{4}{3}\ \-\frac{1}{3}&\frac{1}{3}\end{pmatrix}$, then we can write the two equations above as:
$\mathbf{x}'(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{b}( t)$, which is a first order inhomogeneous linear system, and the general solution is a particular solution plus the general solution of the corresponding homogeneous equation $\mathbf{x}'(t)=\mathbf{A}\mathbf{x}(t)$
(see here).

As our matrix A has constant coefficients, we can find the solution to the homogeneous equation by finding the eigenvalues and eigenvectors; if he have an eigenvector $\mathbf{v}$ corresponding to the eigenvalue $\lambda$, then $\mathbf{x}(t)=Ce^{\lambda{t}}\mathbf{v}$ is a solution to $\mathbf{x}'(t)=\mathbf{A}\mathbf{x}(t)$.

Now, we have
$\det(\mathbf{A}-\lambda\mathbf{I})=0$
$\begin{vmatrix}\frac{5}{3}-\lambda&\frac{4}{3}\\-\frac{1}{3}&\frac{1}{3}-\lambda\end{vmatrix}=0$
$\left(\lambda-\frac{5}{3}\right)\left(\lambda-\frac{1}{3}\right)+\frac{4}{9}=0$
$\lambda^2-2\lambda+1=0$
$(\lambda-1)^2=0$
$\lambda=1$ is a double root of this equation. Looking for eigenvalues, we want $\begin{pmatrix}x\\y\end{pmatrix}$ such that $\begin{pmatrix}x\\y\end{pmatrix}=\begin{pmatrix}\f rac{5}{3}&\frac{4}{3}\\-\frac{1}{3}&\frac{1}{3}\end{pmatrix}\begin{pmatrix }x\\y\end{pmatrix}$, which when multiplied out, both rows give x=-2y, so $\mathbf{v}_1=\begin{pmatrix}2\\-1\end{pmatrix}$ is an eigenvalue.
This gives one solution $\mathbf{x}_1(t)=C_1e^{t}\begin{pmatrix}2\\-1\end{pmatrix}$. There is, however, a second, linearly independent solution (see here). We want to find a vector $\mathbf{v}_2$ such that $\mathbf{v}_1+\mathbf{v}_2=A\mathbf{v}_2$. Letting $\mathbf{v}_2=\begin{pmatrix}x\\y\end{pmatrix}$, we get:
$\begin{pmatrix}x+2\\y-1\end{pmatrix}=\begin{pmatrix}\frac{5}{3}&\frac{4} {3}\\-\frac{1}{3}&\frac{1}{3}\end{pmatrix}\begin{pmatrix }x\\y\end{pmatrix}$, which multiplies out to give x=-2y+3, so $\mathbf{v}_2=\begin{pmatrix}1\\1\end{pmatrix}$ is one such vector. From this (see here), we have the second solution
$\mathbf{x}_2(t)=C_2e^{t}(t\mathbf{v}_1+\mathbf{v}_ 2)$
$\mathbf{x}_2(t)=C_2e^{t}\begin{pmatrix}2t+1\\1-t\end{pmatrix}$,
and the general solution to the homegeneous equation is:
$\mathbf{x}_h(t)=C_1e^{t}\begin{pmatrix}2\\-1\end{pmatrix}+C_2e^{t}\begin{pmatrix}2t+1\\1-t\end{pmatrix}$.

Now, to find a specific solution to the inhomogeneous equation, try a solution of the form
$\mathbf{x}(t)=e^{3t}\begin{pmatrix}a\\b\end{pmatri x}$. Substituting into our equation $\mathbf{x}'(t)=\mathbf{A}\mathbf{x}(t)+\mathbf{b}( t)$, this gives:
$3e^{3t}\begin{pmatrix}a\\b\end{pmatrix}=e^{3t}\beg in{pmatrix}\frac{5}{3}&\frac{4}{3}\\-\frac{1}{3}&\frac{1}{3}\end{pmatrix}\begin{pmatrix }a\\b\end{pmatrix}+\begin{pmatrix}4e^{3t}\\-2e^{3t}\end{pmatrix}$
Multiplying the matrix, adding on the right hand side, and factoring $e^{3t}$ from both sides,
$\begin{pmatrix}3a\\3b\end{pmatrix}=\begin{pmatrix} \frac{5}{3}a+\frac{4}{3}b+4\\-\frac{1}{3}a+\frac{1}{3}b-2\end{pmatrix}$,
so we get a-b=3 and a+8b=-6, which has solution a=2, b=-1. Thus, we have particular solution $\mathbf{x}_p(t)=e^{3t}\begin{pmatrix}2\\-1\end{pmatrix}$.

Add this to the homogeneous solution, and we have:
$\begin{pmatrix}x(t)\\y(t)\end{pmatrix}=C_1e^{t}\be gin{pmatrix}2\\-1\end{pmatrix}+C_2e^{t}\begin{pmatrix}2t+1\\1-t\end{pmatrix}+e^{3t}\begin{pmatrix}2\\-1\end{pmatrix}$. Our initial condition x(0)=3 and y(0)=0 gives us:
$\begin{pmatrix}3\\{0}\end{pmatrix}=C_1\begin{pmatr ix}2\\-1\end{pmatrix}+C_2\begin{pmatrix}1\\1\end{pmatrix} +\begin{pmatrix}2\\-1\end{pmatrix}$, or
$2C_1+C_2=1$ and $C_2-C_1=1$,
so $C_1=0$, $C_2=1$, giving:
$\begin{pmatrix}x(t)\\y(t)\end{pmatrix}=e^{t}\begin {pmatrix}2t+1\\1-t\end{pmatrix}+e^{3t}\begin{pmatrix}2\\-1\end{pmatrix}$
Or separating,
$x(t)=(2t+1)e^t+2e^{3t}$
$y(t)=(1-t)e^t-e^{3t}$.

--Kevin C.