Mathematics: If the wronskian w of f and g is 3e^(4t) and if f(t)=e^(2t), find g(t).

If the wronskian w of f and g is 3e^(4t) and if f(t)=e^(2t),

Ответ:

miacespedes

18.01.2022

Since the Wronskian is

$W(f,g)=fg'-gf'=3e^{4t}$

and

$f(t)=e^{2t}\text{ AND } f'(t)=2e^{2t}$

then

$g'e^{2t}-2ge^{2t}=3e^{4t}$

Divide by $e^{2t}$

$g'-2g=3e^{2t}$

This is a nonhomogenous first order differential equation. We write this in the form

g'+pg=h

There is a really easy way to solve this. We introduce a dummy factor called the integrating factor. Let

$u'(t)=u(t)p
\\
\\ug'+gu'=(gu)'=uh
$

and

$g=\frac{\int uh dt}{u}$

The integrating factor is

$u(t)=e^{\int- 2dt+c}
\\
\\u(t)=e^{-2t+c}$

Then,

$g(t)=\frac{\int u(t)(3e^{2t})dt+k}{u(t)}=\frac{\int e^{-2t+c}3e^{2t}dt+k}{e^{-2t+c}}=\frac{3\int e^{c}dt+k}{e^{-2t+c}}=\frac{3te^c+k}{e^{-2t+c}}$

So

$g(t)=3te^{2t}+ke^{2t}$

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Ответ:

DanteMadrid33

15.03.2023

Mathematics

M = 1/2, (0.4)
because
the slope, m and y-intercept for line that is plotted on the grid below
= 1/2, (0.4)

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If the wronskian w of f and g is 3e^(4t) and if f(t)=e^(2t), find g(t).

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