Internal problem ID [6495]
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]]
Solve \begin {gather*} \boxed {2 y^{\prime \prime } x^{2}-y^{\prime } x +\left (-x^{2}+1\right ) y-x^{3} \cos \relax (x )-\left (\sin ^{2}\relax (x )\right )=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: 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)+sin(x)^2,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{6} x^{2}+\frac {1}{168} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} x \left (1+\frac {1}{10} x^{2}+\frac {1}{360} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+x^{2} \left (\frac {1}{3}+\frac {1}{10} x -\frac {1}{90} x^{3}+\mathrm {O}\left (x^{4}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.856 (sec). Leaf size: 199
AsymptoticDSolveValue[2*x^2*y''[x]-x*y'[x]+(1-x^2)*y[x]==x^3*Cos[x]+Sin[x]^2,y[x],{x,0,5}]
\[ y(x)\to c_2 x \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right )+c_1 \sqrt {x} \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+\sqrt {x} \left (-\frac {x^{11/2}}{396}+\frac {4 x^{9/2}}{45}+\frac {x^{7/2}}{15}-\frac {2 x^{5/2}}{5}-\frac {2 x^{3/2}}{3}\right ) \left (\frac {x^6}{11088}+\frac {x^4}{168}+\frac {x^2}{6}+1\right )+x \left (-\frac {x^6}{168}-\frac {13 x^5}{12600}-\frac {x^4}{12}-\frac {x^3}{18}+\frac {x^2}{2}+x\right ) \left (\frac {x^6}{28080}+\frac {x^4}{360}+\frac {x^2}{10}+1\right ) \]