12.14.26 problem 28

Internal problem ID [1967]
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 : 28
Date solved : Monday, January 27, 2025 at 05:39:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (8+x \right ) y^{\prime \prime }+x \left (3 x +2\right ) y^{\prime }+\left (1+x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 47

Order:=6; 
dsolve(x^2*(8+x)*diff(y(x),x$2)+x*(2+3*x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{{1}/{4}} \left (1-\frac {25}{96} x +\frac {675}{14336} x^{2}-\frac {38025}{5046272} x^{3}+\frac {732615}{645922816} x^{4}-\frac {9230949}{56103010304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_2 \sqrt {x}\, \left (1-\frac {9}{40} x +\frac {5}{128} x^{2}-\frac {245}{39936} x^{3}+\frac {6615}{7241728} x^{4}-\frac {7623}{57933824} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[x^2*(8+x)*D[y[x],{x,2}]+x*(2+3*x)*D[y[x],x]+(1+x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (-\frac {7623 x^5}{57933824}+\frac {6615 x^4}{7241728}-\frac {245 x^3}{39936}+\frac {5 x^2}{128}-\frac {9 x}{40}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {9230949 x^5}{56103010304}+\frac {732615 x^4}{645922816}-\frac {38025 x^3}{5046272}+\frac {675 x^2}{14336}-\frac {25 x}{96}+1\right ) \]