Internal problem ID [1204]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.1 Exercises. Page 318
Problem number: 26.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+2\right ) y^{\prime \prime }+2 x \left (x^{2}+5\right ) y^{\prime }+2 \left (-x^{2}+3\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 49
Order:=6; dsolve(x^2*(2+x^2)*diff(y(x),x$2)+2*x*(x^2+5)*diff(y(x),x)+2*(3-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1+\frac {1}{8} x^{2}+\mathrm {O}\left (x^{6}\right )\right ) x^{2}+\left (2 x^{2}+\frac {1}{4} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \ln \relax (x ) c_{2}+\left (-2-\frac {3}{2} x^{2}-\frac {1}{4} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.015 (sec). Leaf size: 51
AsymptoticDSolveValue[x^2*(2+x^2)*y''[x]+2*x*(x^2+5)*y'[x]+2*(3-x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {x^4+7 x^2+4}{4 x^3}-\frac {\left (x^2+8\right ) \log (x)}{8 x}\right )+c_2 \left (\frac {x}{8}+\frac {1}{x}\right ) \]