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Naya [18.7K]
3 years ago
13

If thewronskian w of f and g is 3e4t,and if f(t) = e2t,find g(t).

Mathematics
1 answer:
ehidna [41]3 years ago
8 0
W(f(x),g(x))=\begin{vmatrix}f(x)&g(x)\\f'(x)&g'(x)\end{vmatrix}=f(x)g'(x)-g(x)f'(x)

We have f(t)=e^{2t}\implies f'(t)=2e^{2t}, so

W(f(t),g(t))=e^{2t}g'(t)-2e^{2t}g(t)=3e^{4t}
\implies e^{-2t}g'(t)-2e^{-2t}g(t)=3
\implies\dfrac{\mathrm d}{\mathrm dt}[e^{-2t}g(t)]=3
\implies e^{-2t}g(t)=\displaystyle\int3\,\mathrm dt
\implies e^{-2t}g(t)=3t+C
\implies g(t)=3te^{2t}+Ce^{2t}

where C is any arbitrary constant.
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