Internal problem ID [1321]
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: 32.
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 +6\right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }+\left (1+2 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 47
Order:=6; dsolve(x^2*(6+x)*diff(y(x),x$2)+x*(11+4*x)*diff(y(x),x)+(1+2*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (1-\frac {10}{63} x +\frac {200}{7371} x^{2}-\frac {17600}{3781323} x^{3}+\frac {3872}{4861701} x^{4}-\frac {921536}{6782072895} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2} x^{\frac {1}{6}}+\left (1-\frac {3}{20} x +\frac {9}{352} x^{2}-\frac {105}{23936} x^{3}+\frac {6615}{8808448} x^{4}-\frac {11907}{92889088} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 90
AsymptoticDSolveValue[x^2*(6+x)*y''[x]+x*(11+4*x)*y'[x]+(1+2*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (-\frac {921536 x^5}{6782072895}+\frac {3872 x^4}{4861701}-\frac {17600 x^3}{3781323}+\frac {200 x^2}{7371}-\frac {10 x}{63}+1\right )}{\sqrt [3]{x}}+\frac {c_2 \left (-\frac {11907 x^5}{92889088}+\frac {6615 x^4}{8808448}-\frac {105 x^3}{23936}+\frac {9 x^2}{352}-\frac {3 x}{20}+1\right )}{\sqrt {x}} \]