30.1.13 problem 13

Internal problem ID [7403]
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 : 13
Date solved : Tuesday, September 30, 2025 at 04:30:41 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=3 x^{2} \left (1+y^{2}\right )^{{3}/{2}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 20

ode:=diff(y(x),x) = 3*x^2*(1+y(x)^2)^(3/2); 
dsolve(ode,y(x), singsol=all);
 

\[ c_1 +x^{3}-\frac {y}{\sqrt {1+y^{2}}} = 0 \]

✓ Mathematica. Time used: 0.168 (sec). Leaf size: 83

ode=D[y[x],x]==3*x^2*(1+y[x]^2)^(3/2); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {i \left (x^3+c_1\right )}{\sqrt {x^6+2 c_1 x^3-1+c_1{}^2}}\\ y(x)&\to \frac {i \left (x^3+c_1\right )}{\sqrt {x^6+2 c_1 x^3-1+c_1{}^2}}\\ y(x)&\to -i\\ y(x)&\to i \end{align*}

✓ Sympy. Time used: 0.662 (sec). Leaf size: 60

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x**2*(y(x)**2 + 1)**(3/2) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \sqrt {- \frac {1}{C_{1}^{2} + 2 C_{1} x^{3} + x^{6} - 1}} \left (- C_{1} - x^{3}\right ), \ y{\left (x \right )} = \sqrt {- \frac {1}{C_{1}^{2} + 2 C_{1} x^{3} + x^{6} - 1}} \left (C_{1} + x^{3}\right )\right ] \]