Internal problem ID [1449]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
III. Exercises 7.7. Page 389
Problem number: 33.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x \left (-2 x^{2}+1\right ) y^{\prime }-4 \left (1-x^{2}\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: 45
Order:=6; dsolve(x^2*diff(y(x),x$2)+x*(1-2*x^2)*diff(y(x),x)-4*(1-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1+\mathrm {O}\left (x^{6}\right )\right )+\ln \relax (x ) \left (288 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (-144-288 x^{2}-216 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 37
AsymptoticDSolveValue[x^2*y''[x]+x*(1-2*x^2)*y'[x]-4*(1-x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 x^2+c_1 \left (\frac {2 x^4+2 x^2+1}{x^2}-2 x^2 \log (x)\right ) \]