Question

Find the general solution to the following system:

dx/dt= x+y
dy/dt=2y

Answer 1

so solving the 2nd equation first I get
\[y=ce ^{2t}\]

Then work for the second one:
\[dx/dt = x + ce ^{2t}\]

\[x = \alpha ce ^{2t}\]
\[x' = 2\alpha ce ^{2t}\]

\[2\alpha ce ^{2t}-\alpha ce ^{2t} = ce ^{2t}\]

Therefore \[\alpha = 1\]
and \[x=ce ^{2t}\]

and the general solution is (ce^2t,ce^2t).

But I think that's wrong.
Anyone shed some light?

Answer 2

There is one solution missing.

Answer 3

In the second equation, you only found the particular solution.

Answer 4

Let me explain what I mean. From the 2nd equation, you found that:
\[y=ce^{2x}\]
And then you substitute that into the first equation, and that gives you:
\[x'=x+ce^{2t} \implies x'-x=ce^{2t}\]
You found the particular solution for this which is:
\[x_p(t)=ce^{2t}\]
The complimentary solution can be found using the auxiliary equation, associated with the homogeneous DE:
\[m-1=0 \implies m=1 \implies x_c(t)=ke^t\]
Therefore the general solutions are:\[x(t)=ce^{2t}+ke^t\]\[y(t)=ce^{2t}\]

Answer 5

Does that makes sense?

Answer 6

Yes, I think so. I'm going to try to find that section of my book and review more as well. I appreciate the help .