54.1.412 problem 423

Internal problem ID [11726]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 423
Date solved : Tuesday, September 30, 2025 at 10:14:43 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.031 (sec). Leaf size: 44

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

\begin{align*} y &= \left (1-\sqrt {2}\right ) x \\ y &= \left (1+\sqrt {2}\right ) x \\ y &= \frac {2 c_1^{2}+2 c_1 x +x^{2}}{2 c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 78

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

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

✓ Sympy. Time used: 2.036 (sec). Leaf size: 17

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

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