64.15.16 problem 16

Internal problem ID [13593]
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 : 16
Date solved : Tuesday, January 28, 2025 at 05:53:32 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x^{2} y^{\prime }-2 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.063 (sec). Leaf size: 45

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

\[ y = c_{1} x^{2} \left (1-\frac {1}{2} x +\frac {3}{20} x^{2}-\frac {1}{30} x^{3}+\frac {1}{168} x^{4}-\frac {1}{1120} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_{2} \left (12-6 x +x^{3}-\frac {1}{2} x^{4}+\frac {3}{20} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x} \]

✓ Solution by Mathematica

Time used: 0.026 (sec). Leaf size: 63

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

\[ y(x)\to c_1 \left (-\frac {x^3}{24}+\frac {x^2}{12}+\frac {1}{x}-\frac {1}{2}\right )+c_2 \left (\frac {x^6}{168}-\frac {x^5}{30}+\frac {3 x^4}{20}-\frac {x^3}{2}+x^2\right ) \]