71.1.145 problem 171 (page 245)

Internal problem ID [19321]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 171 (page 245)
Date solved : Thursday, October 02, 2025 at 04:18:50 PM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 31

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

\begin{align*} y(x)&\to c_1 \cosh \left (\frac {x^2}{2}\right )+i c_2 \sinh \left (\frac {x^2}{2}\right ) \end{align*}

✓ Sympy. Time used: 0.107 (sec). Leaf size: 27

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

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