64.15.18 problem 18

Internal problem ID [13516]
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 : Wednesday, March 05, 2025 at 10:02:59 PM
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*}

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

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

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

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

ode=x^2*D[y[x],{x,2}]-x*D[y[x],x]+3/4*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

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

✓ Sympy. Time used: 0.715 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) - x*Derivative(y(x), x) + 3*y(x)/4,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {3}{2}} + C_{1} \sqrt {x} + O\left (x^{6}\right ) \]