83.18.9 problem 9

Internal problem ID [19140]
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, March 13, 2025 at 01:45:09 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 33

ode:=x+y(x)*diff(y(x),x)^2 = diff(y(x),x)*(1+x*y(x)); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.083 (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: 0.602 (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)
 

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