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 : Sunday, March 30, 2025 at 10:45:16 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 77

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

\begin{align*} y &= \left (2 \sqrt {1+x}\, x -4 \sqrt {1+x}+c_1 \right )^{{1}/{3}} \\ y &= -\frac {\left (\left (2 x -4\right ) \sqrt {1+x}+c_1 \right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2} \\ y &= \frac {\left (\left (2 x -4\right ) \sqrt {1+x}+c_1 \right )^{{1}/{3}} \left (-1+i \sqrt {3}\right )}{2} \\ \end{align*}

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

ode=D[y[x],x]==x/(y[x]^2*Sqrt[1+x]); 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 1.132 (sec). Leaf size: 99

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x/(sqrt(x + 1)*y(x)**2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\left (-1 - \sqrt {3} i\right ) \sqrt [3]{C_{1} + 2 x \sqrt {x + 1} - 4 \sqrt {x + 1}}}{2}, \ y{\left (x \right )} = \frac {\left (-1 + \sqrt {3} i\right ) \sqrt [3]{C_{1} + 2 x \sqrt {x + 1} - 4 \sqrt {x + 1}}}{2}, \ y{\left (x \right )} = \sqrt [3]{C_{1} + 2 x \sqrt {x + 1} - 4 \sqrt {x + 1}}\right ] \]