Internal problem ID [1383]
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
II. Exercises 7.6. Page 374
Problem number: 31.
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}+1\right ) y^{\prime \prime }-x \left (-2 x^{2}+1\right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 45
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(1-2*x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = x \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{2} x^{2}+\frac {3}{8} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {1}{4} x^{2}+\frac {7}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 65
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]-x*(1-2*x^2)*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {3 x^4}{8}-\frac {x^2}{2}+1\right )+c_2 \left (x \left (\frac {7 x^4}{32}-\frac {x^2}{4}\right )+x \left (\frac {3 x^4}{8}-\frac {x^2}{2}+1\right ) \log (x)\right ) \]