MAT244-2014F > Quiz 2

Quiz 2 Problem 1 (night sections)

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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  $W = \det \begin{bmatrix} f(t) \ \ \ g(t) \\fâ€™(t) \ \ gâ€™(t)\end{bmatrix}$

3e^{4t} = f(t)gâ€™(t) - g(t)fâ€™(t)

We know $f(t) = e^{2t}$. Therefore $fâ€™(t) = 2e^{2t}$. Substitute the terms in
$$3e^{4t} = (e^{2t})gâ€™(t) - (2e^{2t})g(t)$$
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}