71.1.76 problem 95 (page 135)

Internal problem ID [19252]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 95 (page 135)
Date solved : Thursday, October 02, 2025 at 03:52:19 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.017 (sec). Leaf size: 24

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

\begin{align*} y &= 2 x \\ y &= 0 \\ y &= \frac {\left (x +c_1 \right )^{2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.288 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to -2 x \sinh ^2\left (\frac {1}{2} (-\log (x)+c_1)\right )\\ y(x)&\to -2 x \sinh ^2\left (\frac {1}{2} (\log (x)+c_1)\right )\\ y(x)&\to 0\\ y(x)&\to 2 x \end{align*}

✓ Sympy. Time used: 1.747 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = x^{2} e^{- C_{1}} + x + \frac {e^{C_{1}}}{4} \]