54.2.7 problem 8

Internal problem ID [8539]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 16. Nonlinear equations. Miscellaneous Exercises. Page 340
Problem number : 8
Date solved : Wednesday, March 05, 2025 at 06:04:12 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.007 (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 {c_{1} +2 x} \\ y &= -\sqrt {c_{1} +2 x} \\ y &= \sqrt {x^{2}+c_{1}} \\ y &= -\sqrt {x^{2}+c_{1}} \\ \end{align*}

✓ Mathematica. Time used: 0.143 (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: 0.943 (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)
 

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