# Toronto Math Forum

## MAT244-2014F => MAT244 Math--Tests => Quiz 2 => Topic started by: Chang Peng (Eddie) Liu on October 01, 2014, 10:39:30 PM

Title: Quiz 2 Problem 1 (night sections)
Post by: Chang Peng (Eddie) Liu on October 01, 2014, 10:39:30 PM
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.
Title: Re: Quiz 2 Problem 1 (night sections)
Post by: Roro Sihui Yap on October 02, 2014, 11:16:18 AM
Consider Wronskian  $W = \det \begin{bmatrix} f(t) \ \ \ g(t) \\fâ€™(t) \ \ gâ€™(t)\end{bmatrix}$
\begin{equation}
3e^{4t} = f(t)gâ€™(t) - g(t)fâ€™(t)
\end{equation}
We know $f(t) = e^{2t}$. Therefore $fâ€™(t) = 2e^{2t}$. Substitute the terms in
\begin{equation} 3e^{4t} = (e^{2t})gâ€™(t) - (2e^{2t})g(t) \end{equation}
Divide all terms by $(e^{2t})$
\begin{gather} 3e^{2t} = gâ€™(t) - 2g(t) \\
gâ€™(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} gâ€™(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}