37.3.4 problem 10.4.8 (d)

Internal problem ID [6410]
Book : Basic Training in Mathematics. By R. Shankar. Plenum Press. NY. 1995
Section : Chapter 10, Differential equations. Section 10.4, ODEs with variable Coefficients. Second order and Homogeneous. page 318
Problem number : 10.4.8 (d)
Date solved : Sunday, March 30, 2025 at 10:54:55 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 27

ode:=x*diff(diff(y(x),x),x)+1/2*diff(y(x),x)+2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sin \left (2 \sqrt {x}\, \sqrt {2}\right )+c_2 \cos \left (2 \sqrt {x}\, \sqrt {2}\right ) \]

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 38

ode=x*D[y[x],{x,2}]+1/2*D[y[x],x]+2*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (2 \sqrt {2} \sqrt {x}\right )+c_2 \sin \left (2 \sqrt {2} \sqrt {x}\right ) \]

✓ Sympy. Time used: 0.194 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + 2*y(x) + Derivative(y(x), x)/2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \sqrt [4]{x} \left (C_{1} J_{\frac {1}{2}}\left (2 \sqrt {2} \sqrt {x}\right ) + C_{2} Y_{\frac {1}{2}}\left (2 \sqrt {2} \sqrt {x}\right )\right ) \]