Internal problem ID [1453]
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: 37.
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 (-x^{2}+5\right ) y^{\prime }-\left (25 x^{2}+7\right ) 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: 35
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(5-x^2)*diff(y(x),x)-(7+25*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{7} \left (1-\frac {6}{5} x^{2}+\frac {7}{5} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-203212800+406425600 x^{2}-609638400 x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 40
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]-x*(5-x^2)*y'[x]-(7+25*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (3 x^3-2 x+\frac {1}{x}\right )+c_2 \left (\frac {7 x^{11}}{5}-\frac {6 x^9}{5}+x^7\right ) \]