Question

Find a particular solution of y"-4y'+4y=e^(2x)(1-3x+6x^2).

Answer 1

I just need to know the initial guess yp. So what should yp be in this problem?

Answer 2

@phi

Answer 3

@amistre64 @mathstudent55 @Directrix

Answer 4

i was never good at guessing, but it might look like something similar to teh right side

Answer 5

Before you attempt to guess a particular solution, you should always determine the characteristic solution first to make sure they don't conflict with your guess solution ("conflict" in the sense that \(y_c\) and \(y_p\) are not linearly independent, that is).

I trust you're familiar with the process, so I'll gloss over the details. You would find the characteristic solution to be
\[y_c=C_1e^{2x}+C_2xe^{2x}\]
Now, given the form of the RHS of the ODE, a decent trial solution might be a same-degree polynomial multiplied by an exponential, i.e. something of the form \(y_p=e^{2x}(a_0+a_1x+a_2x^2)\). However, the first two terms are already covered by the characteristic solution - \(C_1e^{2x}\) would absorb \(a_0e^{2x}\) and \(C_2xe^{2x}\) would absorb \(a_1xe^{2x}\).

To guarantee that \(y_p\) and \(y_c\) are linearly independent, try \({y_p}^*=x^2y_p\), i.e.
\[{y_p}^*=e^{2x}(a_0x^2+a_1x^3+a_2x^4)\]

Answer 6

Thank you, it worked!