Question

Find a particular solution of y"+y=(-4+8x)cosx+(8-4x)sinx.

Answer 1

\[y _{p}=x[(A+Bx)cosx+(C+Dx)sinx]\]

Answer 2

Is my initial guess right above?

Answer 3

The solution to the corresponding homogeneous equation is:
\(\color{#0000ff}{ \displaystyle y_p(x)=c_1\cos(x) + c_2\sin( x) }\)

So, if you were to just write:
\(\color{#0000ff}{ \displaystyle y_p(x)=(A_1x+A_2)\cos(x) +(A_3x+A_4) \sin( x) }\)

Then, you are repeating the homogeneous solution.
\(\color{#0000ff}{ \displaystyle y_p(x)=A_1x\cos(x)+\color{red}{\underline{\color{blue}{A_2\cos(x)}}} +A_3x\sin( x)+\color{red}{\underline{\color{blue}{A_4 \sin( x)}}} }\)

For this reason, (like you did, good job) you add another linear factor \(x\). So, your initial guess is, indeed like you said:

\(\color{#0000ff}{ \displaystyle y_p(x)=\color{green}{\underline{\color{blue}{x}}}(A_1x+A_2)\cos(x) +\color{green}{\underline{\color{blue}{x}}}(A_3x+A_4) \sin( x) }\)

Answer 4

I am using \(A_i\) for coefficients, to track them better, if there are a lot of them, I hope that isn't a problem:)

Now, since this initial guess is a solution to the differential equation (except that you don't know the coefficients yet), you have to plug this initial guess into your differential equation, and find your coefficients.

Answer 5

good luck!

Answer 6

Thank you for the help, it worked!