Second Order Differential Nonhomogeneous Equation

[GogumaDork]

Homework Statement

Use the method of undetermined coefficients to find one solution of
http://img85.imageshack.us/img85/6844/4ab921ad6ba6851cc91401c.png
Note that the method finds a specific solution, not the general one.

Homework Equations

Y = Yc + Yp
Yc = C1e^(r1t)+C2e^(rt) when roots are not the same.

The Attempt at a Solution

For the homogeneous part I used the quadratic equation to get the roots -1 and -4.
Y = Yc + Yp
Yc = C1e^-t +C2e^-4t

I tried Yp = (At^2+Bt+C)*De^(4t) but it doesn't seem correct after solving a bunch of derivatives and plugging it back into y'' + 3y' -4y and solving for constants.

How to determine the Y-particular for this problem?

 

[sagardip]

Hii. Method of undetermined coefficients is sometimes lengthy but it sure does give the correct particular integral. As u hav already written the rough Yp,what u hav to do now is just replace it for y in the original equation and compare the coefficients of (e^4t)t^2,(e^4t)t and e^4t on both sides of the equation and u will get the expression for Yp. There are however other useful and short methods for finding Yp. One is the D Operator method and the other is the variation of parameters method. These are too long for me to describe elaborately here,but u must check these out in the net or any other reference books.Any standard engineering mathematics book would contain these topics.Personally I recommend using the D Operator method as it is very useful when exploited efficiently.

 

[sagardip]

And welcome to Physics Forums...

 

[Dick]

Homework Statement

Use the method of undetermined coefficients to find one solution of
http://img85.imageshack.us/img85/6844/4ab921ad6ba6851cc91401c.png
Note that the method finds a specific solution, not the general one.

Homework Equations

Y = Yc + Yp
Yc = C1e^(r1t)+C2e^(rt) when roots are not the same.

The Attempt at a Solution

For the homogeneous part I used the quadratic equation to get the roots -1 and -4.
Y = Yc + Yp
Yc = C1e^-t +C2e^-4t

I tried Yp = (At^2+Bt+C)*De^(4t) but it doesn't seem correct after solving a bunch of derivatives and plugging it back into y'' + 3y' -4y and solving for constants.

How to determine the Y-particular for this problem?

Can you show what you did and where it went wrong? The numbers aren't pretty but it certainly seems to be working fine for me. You don't need the D parameter. Set it equal to 1. Then I'll get you started. A=(-1/24).

 

[sagardip]

And one more thing, the roots of your auxiliary equation are 1 and -4 respectively.

 

[Dick]

And one more thing, the roots of your auxiliary equation are 1 and -4 respectively.

That would be true. Nice catch. But you don't need them to find the particular solution.