89.32.7 problem 8

Internal problem ID [24967]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 16. Equations of order one and higher degree. Miscellaneous Exercises at page 246
Problem number : 8
Date solved : Thursday, October 02, 2025 at 11:45:07 PM
CAS classification : [_quadrature]

\begin{align*} y^{2} {y^{\prime }}^{2}-\left (1+x \right ) y y^{\prime }+x&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 45

ode:=y(x)^2*diff(y(x),x)^2-y(x)*(1+x)*diff(y(x),x)+x = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \sqrt {2 x +c_1} \\ y &= -\sqrt {2 x +c_1} \\ y &= \sqrt {x^{2}+c_1} \\ y &= -\sqrt {x^{2}+c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.083 (sec). Leaf size: 72

ode=y[x]^2*D[y[x],x]^2-y[x]*(x+1)*D[y[x],x]+x==0; 
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 -\sqrt {x^2+2 c_1}\\ y(x)&\to \sqrt {x^2+2 c_1} \end{align*}

✓ Sympy. Time used: 0.596 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x - (x + 1)*y(x)*Derivative(y(x), x) + y(x)**2*Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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