Internal problem ID [6492]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 24.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _linear, _nonhomogeneous]]
\[ \boxed {2 y^{\prime \prime } x^{2}-y^{\prime } x +\left (1-x^{2}\right ) y-x^{3}-\cos \left (x \right )=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 47
Order:=6; dsolve(2*x^2*diff(y(x), x, x) - x*diff(y(x), x) + (-x^2 + 1)*y(x) = x^3+cos(x),y(x),type='series',x=0);
\[ y \left (x \right ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{6} x^{2}+\frac {1}{10} x^{3}+\frac {5}{504} x^{4}+\frac {1}{360} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.038 (sec). Leaf size: 176
AsymptoticDSolveValue[2*x^2*y''[x]-x*y'[x]+(1-x^2)*y[x]==Cos[x],y[x],{x,0,5}]
\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{3861}+\frac {x^{7/2}}{630}+\frac {4 x^{3/2}}{15}+\frac {2}{\sqrt {x}}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (\frac {37 x^5}{69300}-\frac {x^3}{84}-\frac {x}{3}-\frac {1}{x}\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]