67.3.4 problem 4.3 (d)

Internal problem ID [16325]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 4. SEPARABLE FIRST ORDER EQUATIONS. Additional exercises. page 90
Problem number : 4.3 (d)
Date solved : Thursday, October 02, 2025 at 01:20:35 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.001 (sec). Leaf size: 20

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

\[ y = \frac {x \sqrt {x^{2}+1}}{2}+\frac {\operatorname {arcsinh}\left (x \right )}{2}+c_1 \]

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \frac {1}{2} \left (\text {arcsinh}(x)+\sqrt {x^2+1} x\right )+c_1 \end{align*}

✓ Sympy. Time used: 0.104 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = C_{1} + \frac {x \sqrt {x^{2} + 1}}{2} + \frac {\operatorname {asinh}{\left (x \right )}}{2} \]