Internal problem ID [1334]
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: 45.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+x \left (8 x^{2}+3\right ) y^{\prime }-\left (-4 x^{2}+3\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 33
Order:=6; dsolve(2*x^2*(1+x^2)*diff(y(x),x$2)+x*(3+8*x^2)*diff(y(x),x)-(3-4*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {5}{2}} \left (1-\frac {2}{3} x^{2}+\frac {20}{39} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{4} x^{2}+\frac {5}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 48
AsymptoticDSolveValue[2*x^2*(1+x^2)*y''[x]+x*(3+8*x^2)*y'[x]-(3-4*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {20 x^4}{39}-\frac {2 x^2}{3}+1\right )+\frac {c_2 \left (\frac {5 x^4}{32}-\frac {x^2}{4}+1\right )}{x^{3/2}} \]