5.51 problem 50

Internal problem ID [1025]

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.
ODE order: 1.
ODE degree: 1.

CAS Maple gives this as type [[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]

Solve \begin {gather*} \boxed {2 x \left (y+2 \sqrt {x}\right ) y^{\prime }-\left (y+\sqrt {x}\right )^{2}=0} \end {gather*}

Solution by Maple

Time used: 0.0 (sec). Leaf size: 64

dsolve(2*x*(y(x)+2*sqrt(x))*diff(y(x),x)=(y(x)+sqrt(x))^2,y(x), singsol=all)
 

\begin{align*} y \relax (x ) = \frac {-2 x +\sqrt {\ln \relax (x ) x^{2}-x^{2} c_{1}+4 x^{2}}}{\sqrt {x}} \\ y \relax (x ) = -\frac {2 x +\sqrt {\ln \relax (x ) x^{2}-x^{2} c_{1}+4 x^{2}}}{\sqrt {x}} \\ \end{align*}

Solution by Mathematica

Time used: 0.603 (sec). Leaf size: 68

DSolve[2*x*(y[x]+2*Sqrt[x])*y'[x]==(y[x]+Sqrt[x])^2,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*}