MAT244-2014F > Quiz 2
Quiz 2 Problem 1 (night sections)
(1/1)
Chang Peng (Eddie) Liu:
3.2 #17
If the Wronskian $W$ of $f$ and $g$ is $3e^{4t}$ , and if $f (t) = e^{2t}$ , find $g(t)$.
Can anyone type solution? - V.I.
Roro Sihui Yap:
Consider Wronskian $W = \det \begin{bmatrix} f(t) \ \ \ g(t) \\f’(t) \ \ g’(t)\end{bmatrix}$
\begin{equation}
3e^{4t} = f(t)g’(t) - g(t)f’(t)
\end{equation}
We know $f(t) = e^{2t}$. Therefore $f’(t) = 2e^{2t}$. Substitute the terms in
\begin{equation} 3e^{4t} = (e^{2t})g’(t) - (2e^{2t})g(t) \end{equation}
Divide all terms by $(e^{2t})$
\begin{gather} 3e^{2t} = g’(t) - 2g(t) \\
g’(t) - 2g(t) = 3e^{2t} \end{gather}
We need to find an integrating factor $\mu (t) = e^{\int -2\,dt}= e^{-2t}$ . Multiply all terms by $e^{-2t}$
\begin{gather} g’(t)e^{-2t} - 2g(t)e^{-2t} = 3 \\
[g(t)e^{-2t}]' = 3 \\
g(t)e^{-2t} = 3t + c \\
g(t) = 3te^{2t} + ce^{2t} \end{gather}
Nice—I made minor improvements. V.I.
Navigation
[0] Message Index
Go to full version