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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© BrainMass Inc. brainmass.com September 23, 2018, 4:33 pm ad1c9bdddf - https://brainmass.com/math/basic-algebra/solution-form-power-series-ode-8097
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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)
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) +... ...
A solution is found in the form of a power series for an ODE. Independent solutions are solved by using integration.