I am stuck on this question, I would appreciate any help.
If y1 and y2 are linearly independent solutions of
$\displaystyle ty^{''} +2y^{'} +te^{4t}y =0 $
And if W(y1,y2)(1) = 2, find W(y1,y2)(3)
Thanks for any help.
I assume $\displaystyle t>0$ and in that case write:
$\displaystyle y'' + \frac{2}{t}y' + e^{4t}y=0$
Then $\displaystyle W(y_1,y_2) = a \cdot e^{-\int \frac{2}{t} dt} = a\cdot e^{ - 2\ln t} = \frac{a}{t^2}$.
Since $\displaystyle W(y_1,y_2)(1) = 2 \implies \frac{a}{1^2} = 2 \implies a=2$.
Now you can find $\displaystyle W(y_1,y_2)(3)$