15.19.9 problem 9

Internal problem ID [3293]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 37, page 171
Problem number : 9
Date solved : Tuesday, March 04, 2025 at 04:33:40 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{3}+\left (x +y-2 x y\right ) {y^{\prime }}^{2}-2 y^{\prime } x y \left (x +y\right )&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=diff(y(x),x)^3+(x+y(x)-2*x*y(x))*diff(y(x),x)^2-2*diff(y(x),x)*x*y(x)*(x+y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.054 (sec). Leaf size: 36

ode=D[y[x],x]^3+(x+y[x]-2*x*y[x])*D[y[x],x]^2-2*D[y[x],x]*x*y[x]*(x+y[x])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to c_1 \\ y(x)\to c_1 e^{x^2} \\ y(x)\to -x+c_1 e^{-x}+1 \\ \end{align*}

✓ Sympy. Time used: 0.372 (sec). Leaf size: 22

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*(x + y(x))*y(x)*Derivative(y(x), x) + (-2*x*y(x) + x + y(x))*Derivative(y(x), x)**2 + Derivative(y(x), x)**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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