83.18.12 problem 12

Internal problem ID [19151]
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 (A) at page 53
Problem number : 12
Date solved : Monday, March 31, 2025 at 06:49:50 PM
CAS classification : [_quadrature]

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

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

ode:=diff(y(x),x)^2+x*diff(y(x),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: 0.033 (sec). Leaf size: 32

ode=D[y[x],x]^2+D[y[x],x]*x+D[y[x],x]*y[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: 0.153 (sec). Leaf size: 15

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

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