85.33.39 problem 39

Internal problem ID [22662]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 65
Problem number : 39
Date solved : Thursday, October 02, 2025 at 09:03:12 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} s^{\prime }&=\sqrt {\frac {1-t}{1-s}} \end{align*}

With initial conditions

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

✗ Maple

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

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.492 (sec). Leaf size: 203

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

\begin{align*} s(t)&\to \frac {1}{2} \left (i \sqrt {3} \sqrt [3]{t^3-3 t^2+\left (3-2 i \sqrt {t-1}\right ) t+2 i \sqrt {t-1}-2}-\sqrt [3]{t^3-3 t^2+\left (3-2 i \sqrt {t-1}\right ) t+2 i \sqrt {t-1}-2}+2\right )\\ s(t)&\to \frac {1}{2} \left (i \sqrt {3} \sqrt [3]{t^3-3 t^2+\left (3+2 i \sqrt {t-1}\right ) t-2 i \sqrt {t-1}-2}-\sqrt [3]{t^3-3 t^2+\left (3+2 i \sqrt {t-1}\right ) t-2 i \sqrt {t-1}-2}+2\right ) \end{align*}

✓ Sympy. Time used: 20.230 (sec). Leaf size: 34

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

\[ \sqrt {2} \left (1 - \left (\frac {t - 1}{s{\left (t \right )} - 1}\right )^{\frac {3}{2}}\right ) \left (s{\left (t \right )} - 1\right )^{\frac {3}{2}} = - \sqrt {2} i \]