## anonymous 5 years ago Find the general solution to the following system: dx/dt= x+y dy/dt=2y

1. anonymous

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?

2. anonymous

There is one solution missing.

3. anonymous

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

4. anonymous

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}$

5. anonymous

Does that makes sense?

6. anonymous

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 .