23.2.199 problem 204

Internal problem ID [5554]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 204
Date solved : Tuesday, September 30, 2025 at 12:53:11 PM
CAS classification : [_quadrature]

\begin{align*} y {y^{\prime }}^{2}-\left (1+x y\right ) y^{\prime }+x&=0 \end{align*}

✓ Maple. Time used: 0.005 (sec). Leaf size: 33

ode:=y(x)*diff(y(x),x)^2-(1+x*y(x))*diff(y(x),x)+x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 x +c_1} \\ y &= -\sqrt {2 x +c_1} \\ y &= \frac {x^{2}}{2}+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.058 (sec). Leaf size: 52

ode=y[x] (D[y[x],x])^2-(1+x y[x])D[y[x],x]+x==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.372 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - (x*y(x) + 1)*Derivative(y(x), x) + y(x)*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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