60.3.84 problem 1098

Internal problem ID [11080]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1098
Date solved : Sunday, March 30, 2025 at 07:41:59 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

\begin{align*} x y^{\prime \prime }-y^{\prime }-y a \,x^{3}&=0 \end{align*}

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

ode:=x*diff(diff(y(x),x),x)-diff(y(x),x)-y(x)*a*x^3 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \sinh \left (\frac {x^{2} \sqrt {a}}{2}\right )+c_2 \cosh \left (\frac {x^{2} \sqrt {a}}{2}\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 41

ode=-(a*x^3*y[x]) - D[y[x],x] + x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cosh \left (\frac {\sqrt {a} x^2}{2}\right )+i c_2 \sinh \left (\frac {\sqrt {a} x^2}{2}\right ) \]

✓ Sympy. Time used: 0.190 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(-a*x**3*y(x) + x*Derivative(y(x), (x, 2)) - Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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