83.22.22 problem 22

Internal problem ID [19205]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 22
Date solved : Monday, March 31, 2025 at 06:56:40 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 28

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

\begin{align*} y &= c_1 \,{\mathrm e}^{x} \\ y &= x^{2}+c_1 \\ y &= \ln \left (-{\mathrm e}^{x}-c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 1.023 (sec). Leaf size: 39

ode=D[y[x],x]^3-(y[x]+2*x-Exp[x-y[x]])*D[y[x],x]^2+(2*x*y[x]-2*x*Exp[x-y[x]]-y[x]*Exp[x-y[x]])*D[y[x],x]+2*x*y[x]*Exp[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 x^2+c_1 \\ y(x)\to \log \left (-e^x+c_1\right ) \\ y(x)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.481 (sec). Leaf size: 22

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

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