Varation of Parameters fun diff EQ question, where do i go next?

[mr_coffee]

Varation of Parameters fun! diff EQ question, where do i go next?

This is my first attempt at doing Variation of parameters, didn't go to bad, things cancled out pretty well but now I'm almost done but I'm stuck!
The problem says: Find the solution of y''+15y'+56y = 54*e^(-5t), with y(0) = 8, and y'(0) = 2, y = ?

Here is my work:
http://img157.imageshack.us/img157/6798/s2ly.jpg


http://img157.imageshack.us/img157/6389/s26za.jpg

If you don't follow me or see an error right off the bat, please let me know! Thanks!

 

[HallsofIvy]

Now integrate u₁= -54e^3t and u₂'= 54e^2t to find u₁ and u₂ of course! Once you know those put them into "u₁e^-8t+ u₂e^-7t to get a "specific solution" and add that to the general solution of the homogenous equation.

 

[mr_coffee]

ahh thanks Ivey, I'm alittle confused on what u mean when u said add it to the homogenous equation, the orginal equation wasn't homogenous was set equal to 54e^(-5t), are you saying add it to the r^2+15r+56 = 0? Thanks here is what i got now:
http://img140.imageshack.us/img140/848/lastscan1ia.jpg

 

[HallsofIvy]

Well, that's pretty much the whole idea! If y(t)= C_1e^{-8t}+ C_2e^{-7t} is the general solution to y"+ 15y'+ 56y= 0 and y(t)= 9e^{-5t} satisfies y"+ 15y'+ 56y= 54e^-5t, then
The general solution to y"+ 15y'+56y= 54e^-5t is
y(t)= C_1e^{-8t}+ C_2e^{-7t}+ 9e^{-5t}.

 

[mr_coffee]

Thanks again Ivey!
I don't know if its right because webhw is still down, yay!
but this is what I got:
http://img114.imageshack.us/img114/6418/lastscan3jq.jpg

 

[assyrian_77]

I personally think it's easier to solve it like this: First, as usual, you solve the homogeneous equation which is basically setting the right hand side (RHS) to zero, i.e. y"+ 15y'+56y=0. This will give you y_{h}(t)=C_1e^{-8t}+ C_2e^{-7t}. Next you solve the "special" or "particular" equation by making an ansatz (educated guess) depending on what expression you have on the RHS. In this case I would choose the ansatz y_{p}(t)=Ae^{-5t}. Differentiate this and insert into the original equation gives you a value on A. The total solution is simply y=y_{h}+y_{p}. And then use your initial conditions to get the values of C_1 and C_2.

Sound complicated, but that's how I used to solve these things.

And yes, I too got C_1=-40 and C_2=39.

 

[mr_coffee]

i like ur method a lot better! But how did you make an educational guess of y_{p}(t)=Ae^{-5t} ? I understand why u would guess an e^(t) but how did u konw it should be e^(-5t)? Thanks!

 

[Benny]

You can find out more about that method if you look up 'method of undetermined coefficients.' I find that reduction of order is generally faster because you don't need to memorise specific forms for certain situations.

 

[assyrian_77]

I made the ansatz y_{p}(t)=Ae^{-5t} because the the RHS of the original equation had the exponential term e^{-5t}. More generally, I could have chosen the ansatz y_{p}(t)=Ae^{-5t}+Be^{+5t} but the second term is redundant.