Internal problem ID [1333]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 44.
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}+2\right ) y^{\prime \prime }+x \left (x^{2}+3\right ) y^{\prime }-y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 35
Order:=6; dsolve(x^2*(2+x^2)*diff(y(x),x$2)+x*(3+x^2)*diff(y(x),x)-y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {3}{2}} \left (1-\frac {1}{56} x^{2}+\frac {25}{9856} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{2} x^{2}+\frac {1}{40} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 50
AsymptoticDSolveValue[x^2*(2+x^2)*y''[x]+x*(3+x^2)*y'[x]-y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {25 x^4}{9856}-\frac {x^2}{56}+1\right )+\frac {c_2 \left (\frac {x^4}{40}-\frac {x^2}{2}+1\right )}{x} \]