Help with Wronskian: if W is 3e^(4t), f = e^(2t), find g

[degreeplus]

Hello I need help with this problem:

If the Wronskian W of f and g is \(\displaystyle 3e^{4t}\), and if \(\displaystyle f(t) = e^{2t}\), find g(t).

I have that \(\displaystyle e^{2t}g'(t) - 2e^{2t}g(t) = 3e^{4t}\) and get to like \(\displaystyle g'(t) - 2 g(t) = 3e^{2t}\) but really I feel like I am going in circles, I don't know how to approach this problem.

The answer in the back of the book is \(\displaystyle g(t) = 3te^{2t} + ce^{2t}\)

 

[royhaas]

What is it that you need help with? Do you not know how to solve a first order equation?

 

[Deleted member 4993]

Hello I need help with this problem:

If the Wronskian W of f and g is \(\displaystyle 3e^{4t}\), and if \(\displaystyle f(t) = e^{2t}\), find g(t).

I have that \(\displaystyle e^{2t}g'(t) - 2e^{2t}g(t) = 3e^{4t}\) and get to like \(\displaystyle g'(t) - 2 g(t) = 3e^{2t}\) but really I feel like I am going in circles, I don't know how to approach this problem.

The answer in the back of the book is \(\displaystyle g(t) = 3te^{2t} + ce^{2t}\)

Have dealt with "integrating factor" in first order ODE?

If your DE is:

\(\displaystyle y'\, + \, y* p(x) = q(x)\)

then the integrating factor is

\(\displaystyle e^{\int p\cdot dx}\)

multiplying both sides of the equation by the factor and integrating - you get

\(\displaystyle y \cdot e^{\int p\cdot dx} = \,\int q \cdot e^{\int p\cdot dx} dx\)

Follow this method....