54.3.121 problem 1135

Internal problem ID [12416]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1135
Date solved : Wednesday, October 01, 2025 at 01:44:39 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

ode:=4*x*diff(diff(y(x),x),x)+2*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sinh \left (\sqrt {x}\right )+c_2 \cosh \left (\sqrt {x}\right ) \]

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 27

ode=-y[x] + 2*D[y[x],x] + 4*x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 \cosh \left (\sqrt {x}\right )+i c_2 \sinh \left (\sqrt {x}\right ) \end{align*}

✓ Sympy. Time used: 0.101 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(4*x*Derivative(y(x), (x, 2)) - y(x) + 2*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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