Finding particular solution to differential equations

[lillybeans]

There are two questions that I am trying to solve on web assignment. The goal is to find a general form of a particular solution to each ODE. The question asks me to represent all constants in the solution using "P,Q,R,S,T..etc.", in that order.

1. y'''-9y''+14y'=x²
2. y''-9y'+14y=x²e^4x

For the first one I wrote:
y_p=Px³+Qx²+Rx

Second one:
y_p=(Px²+Qx+R)e^4x

Neither of the answers are correct, according to the computer. Where did I go wrong?

Thanks.

 

[STEMucator]

The answers are not correct because you're not solving the equations correctly... First do you understand that your equations are non-homogenous?

 

[lillybeans]

The answers are not correct because you're not solving the equations correctly... First do you understand that your equations are non-homogenous?

Yes, of course, that's why I am asked to find a particular solution. Where did I go wrong? please let me know!

 

[STEMucator]

Okay, so take for example your second question. Notice that your derivatives and original function have to add up to x²e^4x?

So your general solution must have the form : y = A(x²e^4x) where A is some constant number.

Now take the required derivatives and see if they satisfy your equation. ( You'll get a bunch of constants which you have to add together and solve for A for the particular solution you want ).

 

[lillybeans]

Okay, so take for example your second question. Notice that your derivatives and original function have to add up to x²e^4x?

So your general solution must have the form : y = A(x²e^4x) where A is some constant number.

Now take the required derivatives and see if they satisfy your equation. ( You'll get a bunch of constants which you have to add together and solve for A for the particular solution you want ).

I don't quite understand why your guess for the particular solution has the form Ax²e^4x as opposed to (Ax²+Bx+C)e^4x. Shouldn't you use the general form of the second degree polynomial?

 

[STEMucator]

A non-homogenous equation with constant coefficients has the general solution y = Ag(t) where A is a constant.

 

[lillybeans]

A non-homogenous equation with constant coefficients has the general solution y = Ag(t) where A is a constant.

Then suppose for y''-9y'+14y my y_p=Ax²e^4x

Then

y'=2Axe^4x+4Ax²e^4x
y''=2Ae^4x+8Axe^4x+8Axe^4x+16Ax²e^4x

When you plug this all back into the original equation, equate the coefficients to solve for A, it doesn't work. Need more than just one constant.