Internal problem ID [4413]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson
2018.
Section: Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises. page
46
Problem number: 10.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {x}{y^{2} \sqrt {x +1}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 123
dsolve(diff(y(x),x)=x/(y(x)^2*sqrt(1+x)),y(x), singsol=all)
\begin{align*} y \relax (x ) = \left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1}\right )^{\frac {1}{3}} \\ y \relax (x ) = -\frac {\left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1}\right )^{\frac {1}{3}}}{2}-\frac {i \sqrt {3}\, \left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1}\right )^{\frac {1}{3}}}{2} \\ y \relax (x ) = -\frac {\left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1}\right )^{\frac {1}{3}}}{2}+\frac {i \sqrt {3}\, \left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1}\right )^{\frac {1}{3}}}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 2.285 (sec). Leaf size: 110
DSolve[y'[x]==x/(y[x]^2*Sqrt[1+x]),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{2 \sqrt {x+1} x-4 \sqrt {x+1}+3 c_1} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{2 \sqrt {x+1} x-4 \sqrt {x+1}+3 c_1} \\ y(x)\to (-1)^{2/3} \sqrt [3]{2 \sqrt {x+1} x-4 \sqrt {x+1}+3 c_1} \\ \end{align*}