6.5.51 problem 50

Internal problem ID [1675]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 2, First order equations. Transformation of Nonlinear Equations into Separable Equations. Section 2.4 Page 68
Problem number : 50
Date solved : Tuesday, September 30, 2025 at 04:58:46 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 50

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

\begin{align*} y &= \frac {-2 x +\sqrt {x^{2} \left (\ln \left (x \right )-c_1 +4\right )}}{\sqrt {x}} \\ y &= -\frac {2 x +\sqrt {x^{2} \left (\ln \left (x \right )-c_1 +4\right )}}{\sqrt {x}} \\ \end{align*}

✓ Mathematica. Time used: 0.404 (sec). Leaf size: 68

ode=2*x*(y[x]+2*Sqrt[x])*D[y[x],x]==(y[x]+Sqrt[x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -2 \sqrt {x}-\sqrt {\frac {1}{x^2}} x \sqrt {x (\log (x)+4+c_1)}\\ y(x)&\to -2 \sqrt {x}+\sqrt {\frac {1}{x^2}} x \sqrt {x (\log (x)+4+c_1)} \end{align*}

✓ Sympy. Time used: 19.437 (sec). Leaf size: 36

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

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