# Find a solution in the form of a power series for an ODE

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Find a solution in the form of a power series for the equation

y" - 2*x*y' = 0

(ie find 2 linearly independent solutions y1(x) and y2(x)).

After doing that, note that the equation can also be solved directly by integration:

y"/y' = 2x

ln(y') = x^2 + c1

y' = ke^(x^2) k=e^c1

y = k* integral((e^(t^2))dt + C) from 0 to x

Thus, one of your power series solutions gives and explicit form for the integral:

integral(e^(t^2)dt) from 0 to x

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Solution. Assume that a solution of the ODE (in form of power series) is as follows.

y(x)=a(0)+a(1)x+a(2) +a(3) +...+a(n) +... (1)

where a(i), i=0,1,2,... are coefficients.

Then by (1) we have,

y'(x)=a(1)+2*a(2)x+3*a(3) +...+n*a(n) +(n+1)*a(n+1) +... (2)

and

y''(x)=2*a(2)+6*a(3)x+...+n(n-1)*a(n) +(n+1)n* a(n+1) +(n+2)(n+1)*a(n+2) +... ...

#### Solution Summary

A solution is found in the form of a power series for an ODE. Independent solutions are solved by using integration.