77.18.9 problem 9

Internal problem ID [20504]
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 : 9
Date solved : Thursday, October 02, 2025 at 06:03:40 PM
CAS classification : [_quadrature]

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

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

ode:=x+y(x)*diff(y(x),x)^2 = diff(y(x),x)*(x*y(x)+1); 
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: 0.057 (sec). Leaf size: 52

ode=x+y[x]*D[y[x],x]^2==D[y[x],x]*(1+x*y[x]); 
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: 0.380 (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)
 

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