77.1.76 problem 95 (page 135)

Internal problem ID [17895]
Book : V.V. Stepanov, A course of differential equations (in Russian), GIFML. Moscow (1958)
Section : All content
Problem number : 95 (page 135)
Date solved : Monday, March 31, 2025 at 04:49:01 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.039 (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: 2.279 (sec). Leaf size: 75

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 x \left (2 \cosh (-\log (x)+c_1)-\sqrt {3} \sinh (-\log (x)+c_1)+1\right ) \\ y(x)\to x \left (2 \cosh (\log (x)+c_1)-\sqrt {3} \sinh (\log (x)+c_1)+1\right ) \\ y(x)\to 0 \\ y(x)\to 2 x \\ \end{align*}

✓ Sympy. Time used: 2.604 (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)
 

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