Internal problem ID [6476]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }-x y^{\prime }+\left (1-x^{2}\right ) y-x^{4}=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.02 (sec). Leaf size: 43
Order:=6; dsolve(2*x^2*diff(y(x), x$2) - x*diff(y(x), x) + (1-x^2 )*y(x) = x^4,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+x^{4} \left (\frac {1}{21}+\mathrm {O}\left (x^{2}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.026 (sec). Leaf size: 150
AsymptoticDSolveValue[2*x^2*y''[x] - x*y'[x] + (1-x^2 )*y[x] ==x^4,y[x],{x,0,5}]
\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{55}-\frac {2 x^{7/2}}{7}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {x^5}{30}+\frac {x^3}{3}\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]