73.1.4 problem 1 (iv)

Internal problem ID [19777]
Book : Elementary Differential Equations. By R.L.E. Schwarzenberger. Chapman and Hall. London. First Edition (1969)
Section : Chapter 3. Solutions of first-order equations. Exercises at page 47
Problem number : 1 (iv)
Date solved : Thursday, October 02, 2025 at 04:43:17 PM
CAS classification : [_quadrature]

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

With initial conditions

\begin{align*} x \left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 15

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

\[ x = \operatorname {arcsinh}\left (t \right )-\ln \left (1+\sqrt {2}\right ) \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 12

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

\begin{align*} x(t)&\to \text {arcsinh}(t)-\text {arcsinh}(1) \end{align*}

✓ Sympy. Time used: 0.093 (sec). Leaf size: 14

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

\[ x{\left (t \right )} = \operatorname {asinh}{\left (t \right )} - \log {\left (1 + \sqrt {2} \right )} \]