find the general power series solution to the following differentialfind the general power series solution to the following differential equation up to the x^5 term of the power series solution:...
find the general power series solution to the following differential
We start by assuming a power series solution to the differential equation. That is, let `y=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+a_5x^5+cdots`
Now differentiating, we get
and again to get
where the dots represent higher order terms that we can ignore.
Combining both of these into the left side of the differential equation, we get
And now compare powers of x to zero to solve for the coefficients.
We don't need to go any further since the question is only asking up to `a_5` .
This means that `a_2=a_4=0` , `a_0` and `a_1` are arbitrary, `a_3=1/12 a_1` and `a_5=3/40 a_3=1/160 a_1` .
Therefore, up to the `x^5` term, the power series solution is