Internal problem ID [1454]
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: 38.
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}+5\right ) y^{\prime }-21 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.011 (sec). Leaf size: 35
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)+x*(5+2*x^2)*diff(y(x),x)-21*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{3} \left (1-\frac {1}{2} x^{2}+\frac {15}{56} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (-1316818944000-3456649728000 x^{2}-2880541440000 x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{7}} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 46
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]+x*(5+2*x^2)*y'[x]-21*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{x^7}+\frac {21}{8 x^5}+\frac {35}{16 x^3}\right )+c_2 \left (\frac {15 x^7}{56}-\frac {x^5}{2}+x^3\right ) \]