12.14.30 problem 32

Internal problem ID [1971]
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
Date solved : Monday, January 27, 2025 at 05:39:26 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (6+x \right ) y^{\prime \prime }+x \left (11+4 x \right ) y^{\prime }+\left (2 x +1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 48

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);
 

\[ y = \frac {c_2 \left (1-\frac {10}{63} x +\frac {200}{7371} x^{2}-\frac {17600}{3781323} x^{3}+\frac {3872}{4861701} x^{4}-\frac {921536}{6782072895} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{{1}/{6}}+c_1 \left (1-\frac {3}{20} x +\frac {9}{352} x^{2}-\frac {105}{23936} x^{3}+\frac {6615}{8808448} x^{4}-\frac {11907}{92889088} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.006 (sec). Leaf size: 90

AsymptoticDSolveValue[x^2*(6+x)*D[y[x],{x,2}]+x*(11+4*x)*D[y[x],x]+(1+2*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {921536 x^5}{6782072895}+\frac {3872 x^4}{4861701}-\frac {17600 x^3}{3781323}+\frac {200 x^2}{7371}-\frac {10 x}{63}+1\right )}{\sqrt [3]{x}}+\frac {c_2 \left (-\frac {11907 x^5}{92889088}+\frac {6615 x^4}{8808448}-\frac {105 x^3}{23936}+\frac {9 x^2}{352}-\frac {3 x}{20}+1\right )}{\sqrt {x}} \]