56.13.1 problem Ex 1

Internal problem ID [14157]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IV, differential equations of the first order and higher degree than the first. Article 24. Equations solvable for \(p\). Page 49
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:17:50 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

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

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

✓ Mathematica. Time used: 0.023 (sec). Leaf size: 32

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

\begin{align*} y(x)&\to c_1 e^{-x}\\ y(x)&\to -\frac {x^2}{2}+c_1\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.091 (sec). Leaf size: 15

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

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