Internal problem ID [1456]
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: 40.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+4 x \left (x^{2}+2\right ) y^{\prime }-\left (x^{2}+15\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 35
Order:=6; dsolve(4*x^2*(1+x^2)*diff(y(x),x$2)+4*x*(2+x^2)*diff(y(x),x)-(15+x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{4} \left (1-\frac {1}{6} x^{2}+\frac {1}{16} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (-144-216 x^{2}-54 x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {5}{2}}} \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 58
AsymptoticDSolveValue[4*x^2*(1+x^2)*y''[x]+4*x*(2+x^2)*y'[x]-(15+x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {3 x^{3/2}}{8}+\frac {1}{x^{5/2}}+\frac {3}{2 \sqrt {x}}\right )+c_2 \left (\frac {x^{11/2}}{16}-\frac {x^{7/2}}{6}+x^{3/2}\right ) \]