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Solve the differential equation below, using power series$$ 2y'' - 6x*y' = 7 $$Hi sorry to bother you guys but I'm having trouble answering because I'm not good at solving this equation,

engineering student need help
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    Welcome to the forum! Have you tried to solve this yourself? I'm asking because if the answer is yes, then it could be that you're just a small step away from finishing the solution. So please show your own work. – Matti P. Mar 31 '21 at 10:13
  • Replace $y$ by $a_0+a_1x+a_2x^2+....$ (or better $\Sigma_{n=0}^{\infty}a_nx^n$) in the two places where it occurs, then identify coefficients of the same $x^n$ in LHS and RHS. This will give you a recurrence relationship. – Jean Marie Mar 31 '21 at 10:22
  • Thank you but can you please elaborate more cause I'm very slow at learning – engineering student need help Mar 31 '21 at 10:23
  • $2(a_0+a_1x+a_2x^2+a_3x^3+....)''-6x(a_0+a_1x+a_2x^2+....)'=7 \iff 2(2a_2+6a_3x+...)-6x(a_1+2_2x+....)=7 \iff 4a_2=7, 12a_3-6a_1=0, $ etc. giving you $a_2=7/4, a_3=\frac12 a_1...$, etc. – Jean Marie Mar 31 '21 at 10:28
  • thank you! how about this one? Determine the power series expansion with at least three non-zero terms for [3cos(x)+5(e^x)^2]. – engineering student need help Mar 31 '21 at 10:39

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