Question

y' - 2y = e^3t, y(0) = 3, Find the general solution and solve for the IVP. I have no idea how to solve non homogeneous differential equations

Answer 1

To solve non homogeneous you need to find the general solution of the homogeneous part

y'-2y=0
which would be Yc

and then the particular solution (Yp) for the right hand side

=e^3t

the general solution would be
y=Yc+Yp

Answer 2

The correct answer is e^3t + 2e^2t. I understand how to come up with this but when solving for the homogeneous part I get y=(e^(t^2)/2)/2. What am I doing wrong here?

Answer 3

If you use the characteristic equation, how would your answer be?

Answer 4

r-2=0 right?

Answer 5

I assumed that y' = 2y(t) and then got dy/2y =1dt. DOes the 2 go with the t rather then the y

Answer 6

I have never been told to use r as anything before

Answer 7

http://www.wolframalpha.com/input/?i=y%27+-+2y+%3D+e^%283t%29%2C+y%280%29+%3D+3

Answer 8

You can get a step by step soln there

Answer 9

Okay thank you!

Answer 10

Also for 2nd order how would you solve y"-y'=0.

Answer 11

nevermind sorry I'll just use that website

Answer 12

I learned using characteristic equations

where you substitute y''=r^2, y'=r and then you factor which will give you zeros

y''-y'=0
r^2-r=0
r(r-1)=0
r=0 r=1

general soln
\[y=C _{1}+C _{2}e ^{x}\]