Internal problem ID [6479]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } x^{2}-y^{\prime } x +\left (-x^{2}+1\right ) y-x \sin \relax (x )=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 45
Order:=6; dsolve(2*x^2*diff(y(x), x$2) - x*diff(y(x), x) + (1-x^2 )*y(x) = x*sin(x),y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{126} x^{2}+\mathrm {O}\left (x^{4}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.034 (sec). Leaf size: 167
AsymptoticDSolveValue[2*x^2*y''[x] - x*y'[x] + (1-x^2 )*y[x] ==x*sin(x),y[x],{x,0,5}]
\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right ) \left (-\frac {x^{11/2} \sin }{1980}-\frac {1}{35} x^{7/2} \sin -\frac {2}{3} x^{3/2} \sin \right )+x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \left (\frac {x^5 \sin }{840}+\frac {x^3 \sin }{18}+x \sin \right ) \]