63.1.16 problem 16

Internal problem ID [15456]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 16
Date solved : Thursday, October 02, 2025 at 10:15:05 AM
CAS classification : [_separable]

\begin{align*} 1+s^{2}-\sqrt {t}\, s^{\prime }&=0 \end{align*}

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

ode:=1+s(t)^2-t^(1/2)*diff(s(t),t) = 0; 
dsolve(ode,s(t), singsol=all);
 

\[ s = \tan \left (2 \sqrt {t}+c_1 \right ) \]

✓ Mathematica. Time used: 0.138 (sec). Leaf size: 47

ode=(1+s[t]^2)-Sqrt[t]*D[s[t],t]==0; 
ic={}; 
DSolve[{ode,ic},s[t],t,IncludeSingularSolutions->True]
 

\begin{align*} s(t)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [2 \sqrt {t}+c_1\right ]\\ s(t)&\to -i\\ s(t)&\to i \end{align*}

✓ Sympy. Time used: 0.186 (sec). Leaf size: 12

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

\[ s{\left (t \right )} = \tan {\left (C_{1} + 2 \sqrt {t} \right )} \]