How can I solve this without using integrating factor?

[ktklam9]

Let the Wronskian between the functions f and g to be 3e^{4t}, if f(t) = e^{2t}, then what is g(t)?

So the Wronskian setup is pretty easy

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

f = e^{2t}
f' = 2e^{2t}

So plugging it in I would get:

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

Which results in

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

How can I solve for g without using integrating factor? Is it even possible? Thanks :)

 

[HallsofIvy]

It's always possible to solve "some other way" but often much more difficult.

This particular example, however, is a "linear equation with constant coefficients" which has a fairly simple solution method. Because it is linear, we can add two solutions to get a third so start by looking at g'- 2g= 0.

g'= 2g give dg/g= 2dt and, integrating ln(g)= 2t+ c. Taking the exponential of both sides, g(t)= e^{2t+ c}= e^{2t}e^c= Ce^{2t} where C is defined as e^c.

Now, we can use a method called "variation of parameters" because we allow that "C" in the previous solution to be a variable: let g= v(t)e^{2t}. Then g'= v'(t)e^{2t}+ 2v(t)e^{2t} so the equation becomes g'- 2g= v'(t)e^{2t}+ 2v(t)e^{2t}- 2v(t)e^{2t}= v'(t)e^{2t}= 3e^{2t}. We can cancel the "e^{2t}" terms to get v'(t)= 3 and, integrating, v(t)= 3t+ C. That gives the solution g(t)= v(t)e^{2t}= 3te^{2t}+ Ce^{2t} where "C" can be any number.