38.5.73 problem 62 (a)

Internal problem ID [8421]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 62 (a)
Date solved : Tuesday, September 30, 2025 at 05:36:19 PM
CAS classification : [_quadrature]

\begin{align*} u^{\prime }&=a \sqrt {1+u^{2}} \end{align*}

With initial conditions

\begin{align*} u \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.041 (sec). Leaf size: 8

ode:=diff(u(x),x) = a*(1+u(x)^2)^(1/2); 
ic:=[u(0) = 0]; 
dsolve([ode,op(ic)],u(x), singsol=all);
 

\[ u = \sinh \left (x a \right ) \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 9

ode=D[u[x],x]== a*Sqrt[1+u[x]^2]; 
ic={u[0]==0}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to \sinh (a x) \end{align*}

✓ Sympy. Time used: 0.187 (sec). Leaf size: 7

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

\[ u{\left (x \right )} = \sinh {\left (a x \right )} \]