36.1.10 problem 10

Internal problem ID [6265]
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
Date solved : Monday, January 27, 2025 at 01:50:32 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {x}{y^{2} \sqrt {1+x}} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 77

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

\begin{align*} y &= \left (2 \sqrt {x +1}\, x -4 \sqrt {x +1}+c_{1} \right )^{{1}/{3}} \\ y &= -\frac {\left (\left (2 x -4\right ) \sqrt {x +1}+c_{1} \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (\left (2 x -4\right ) \sqrt {x +1}+c_{1} \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.008 (sec). Leaf size: 110

DSolve[D[y[x],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*}