Internal problem ID [1328]
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: 39.
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}+2\right ) y^{\prime \prime }-x \left (-7 x^{2}+12\right ) y^{\prime }+\left (3 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: 0.012 (sec). Leaf size: 35
Order:=6; dsolve(2*x^2*(2+x^2)*diff(y(x),x$2)-x*(12-7*x^2)*diff(y(x),x)+(7+3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (x^{3} \left (1-\frac {9}{8} x^{2}+\frac {117}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{1}+\left (12+9 x^{2}-\frac {63}{4} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 58
AsymptoticDSolveValue[2*x^2*(2+x^2)*y''[x]-x*(12-7*x^2)*y'[x]+(7+3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (-\frac {21 x^{9/2}}{16}+\frac {3 x^{5/2}}{4}+\sqrt {x}\right )+c_2 \left (\frac {117 x^{15/2}}{128}-\frac {9 x^{11/2}}{8}+x^{7/2}\right ) \]