Internal problem ID [6516]
Book: Own collection of miscellaneous problems
Section: section 4.0
Problem number: 45.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+4 y^{\prime } x +\left (x^{2}+2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 35
Order:=6; dsolve(x^2*diff(y(x), x, x) + 4*x*diff(y(x), x) + (x^2+2)*y(x) = 0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {1}{6} x^{2}+\frac {1}{120} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) x +c_{2} \left (1-\frac {1}{2} x^{2}+\frac {1}{24} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 40
AsymptoticDSolveValue[x^2*y''[x]+4*x*y'[x]+(x^2+2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^3}{120}-\frac {x}{6}+\frac {1}{x}\right )+c_1 \left (\frac {x^2}{24}+\frac {1}{x^2}-\frac {1}{2}\right ) \]