49.19.1 problem 1(i)

Internal problem ID [7721]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number : 1(i)
Date solved : Monday, January 27, 2025 at 03:10:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 2 x^{2} y^{\prime \prime }+\left (x^{2}+5 x \right ) y^{\prime }+\left (x^{2}-2\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 56

Order:=8; 
dsolve(2*x^2*diff(y(x),x$2)+(5*x+x^2)*diff(y(x),x)+(x^2-2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{2} x^{{5}/{2}} \left (1-\frac {1}{14} x -\frac {25}{504} x^{2}+\frac {197}{33264} x^{3}+\frac {1921}{3459456} x^{4}-\frac {11653}{103783680} x^{5}+\frac {12923}{21171870720} x^{6}+\frac {917285}{1126343522304} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{1} \left (1-\frac {2}{3} x +\frac {5}{6} x^{2}+\frac {2}{9} x^{3}-\frac {19}{216} x^{4}-\frac {1}{540} x^{5}+\frac {101}{45360} x^{6}-\frac {4}{35721} x^{7}+\operatorname {O}\left (x^{8}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 116

AsymptoticDSolveValue[2*x^2*D[y[x],{x,2}]+(5*x+x^2)*D[y[x],x]+(x^2-2)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {917285 x^7}{1126343522304}+\frac {12923 x^6}{21171870720}-\frac {11653 x^5}{103783680}+\frac {1921 x^4}{3459456}+\frac {197 x^3}{33264}-\frac {25 x^2}{504}-\frac {x}{14}+1\right )+\frac {c_2 \left (-\frac {4 x^7}{35721}+\frac {101 x^6}{45360}-\frac {x^5}{540}-\frac {19 x^4}{216}+\frac {2 x^3}{9}+\frac {5 x^2}{6}-\frac {2 x}{3}+1\right )}{x^2} \]