Find the general solution?

Question

Find the general solution of y'+3y=t+e^(-2t).

Answer 1

Multiply by an integrating factor μ satisfying dμ/dt = 3μ.

μ dy/dt + 3μ y = μ(t + e^-2t)

μ dy/dt + y dμ/dt = μ(t + e^-2t)

μy = ∫ μ(t + e^-2t) dt

y = (1/μ)∫ μ(t + e^-2t) dt

Now find μ.

dμ/dt = 3μ

∫dμ/μ = ∫3dt

ln |μ| = 3t + C

μ = Ae^3t

Plug it in. Note that the A's will cancel.

y = (1/μ)∫ μ(t + e^-2t) dt = (1/(Ae^3t)) ∫ Ae^3t(t + e^-2t) dt = e^-3t (∫ te^3t dt + ∫ e^t dt)

The left integral can be solved with integration by parts:

u = t, du = dt, dv = e^3t dt, v = e^3t/3

∫ te^3t dt= te^3t/3 - (1/3)∫ e^3t dt = te^3t/3 - e^3t/9 + C

Thus you have

y = e^-3t(te^3t/3 - e^3t/9 + e^t + C) = t/3 - 1/9 + e^-2t + Ce^-3t

Check:

d/dt (t/3 - 1/9 + e^-2t + Ce^-3t) + 3 (t/3 - 1/9 + e^-2t + Ce^-3t)

= 1/3 - 0 - 2e^-2t - 3Ce^-3t + t - 1/3 + 3e^-2t + 3Ce^-3t

= t + e^-2t

Answer 2

These problems are almost always done by "guess and check." That is, find what general form you think the solution will take, then find constants that make it work out. After a little bit of practice, what the right general solution should be (where you should start your "guess and check" work) will become more intuitive and easier to figure out. In this case, as in most, there are two parts to the problem solving process.

The first is to find any solution that works. You will want to try

y = a + bt + ce^(-2t).

Plug this function in for y in the differential equation that you were given and find the values of the constants a, b, and c that make it work. (In this case, there will be values of a, b, and c that make y a solution to the differential equation.)

Now that you have a particular solution, here is step two. Imagine that your friend has some other particular solution, let's call it z. Let w = y - z.

Then w' + 3w = (y-z)' + 3(y-z) = y' - z' + 3y -3z = (y' + 3y) - (z' - 3z) = (t + e^-2t) - (t + e^-2t) = 0.

So we know that w satisfies the differential equation w' + 3w = 0. Put differently, since z = y - w, we know that your friend's solution must be your solution plus some solution to the differential equation w' + 3w = 0.

Now, convince yourself that any function w satisfying this differential equation will work. Then if you can find the general form for functions w satisfying w' + 3w = 0, (try w = d*e^(-3t) for any constant d; can you see why this works?) you will be nearly done.

Putting this all together, you should find that your general solution has the form

y = a + bt + c*e^(-2t) + d*e^(-3t),

where a, b, and c are constants whose values you will have to find, and d can be any constant.