64.15.1 problem 1

Internal problem ID [13578]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 6, Series solutions of linear differential equations. Section 6.2 (Frobenius). Exercises page 251
Problem number : 1
Date solved : Tuesday, January 28, 2025 at 05:53:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.043 (sec). Leaf size: 44

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

\[ y = c_{1} x^{{5}/{3}} \left (1+\frac {17}{36} x +\frac {1241}{7128} x^{2}+\frac {80665}{1347192} x^{3}+\frac {972725}{48498912} x^{4}+\frac {5797441}{872980416} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-\frac {1}{2} x -\frac {1}{2} x^{2}-\frac {5}{24} x^{3}-\frac {25}{336} x^{4}-\frac {17}{672} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.012 (sec). Leaf size: 85

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

\[ y(x)\to c_2 \left (-\frac {17 x^5}{672}-\frac {25 x^4}{336}-\frac {5 x^3}{24}-\frac {x^2}{2}-\frac {x}{2}+1\right )+c_1 \left (\frac {5797441 x^5}{872980416}+\frac {972725 x^4}{48498912}+\frac {80665 x^3}{1347192}+\frac {1241 x^2}{7128}+\frac {17 x}{36}+1\right ) x^{5/3} \]