64.15.18 problem 18

Internal problem ID [13595]
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 : 18
Date solved : Tuesday, January 28, 2025 at 05:53:34 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x^{2} y^{\prime \prime }-x y^{\prime }+\frac {3 y}{4}&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 18

Order:=6; 
dsolve(x^2*diff(y(x),x$2)-x*diff(y(x),x)+3/4*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (c_{1} x +c_{2} \right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 20

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]-x*D[y[x],x]+3/4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 x^{3/2}+c_1 \sqrt {x} \]