85.33.76 problem 77

Internal problem ID [22699]
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 : 77
Date solved : Thursday, October 02, 2025 at 09:11:00 PM
CAS classification : [_quadrature]

\begin{align*} r^{3} r^{\prime }&=\sqrt {a^{8}-r^{8}} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 28

ode:=r(t)^3*diff(r(t),t) = (a^8-r(t)^8)^(1/2); 
dsolve(ode,r(t), singsol=all);
 

\[ t -\int _{}^{r}\frac {\textit {\_a}^{3}}{\sqrt {-\textit {\_a}^{8}+a^{8}}}d \textit {\_a} +c_1 = 0 \]

✓ Mathematica. Time used: 3.406 (sec). Leaf size: 346

ode=r[t]^3*D[r[t],t]==Sqrt[a^8-r[t]^8]; 
ic={}; 
DSolve[{ode,ic},r[t],t,IncludeSingularSolutions->True]
 

\begin{align*} r(t)&\to -\frac {a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to -\frac {i a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to \frac {i a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to \frac {a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to -\frac {\sqrt [4]{-1} a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to \frac {\sqrt [4]{-1} a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to -\frac {(-1)^{3/4} a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to \frac {(-1)^{3/4} a \sqrt [4]{\tan (4 (t+c_1))}}{\sqrt [8]{\sec ^2(4 (t+c_1))}}\\ r(t)&\to -a\\ r(t)&\to -i a\\ r(t)&\to i a\\ r(t)&\to a\\ r(t)&\to -\sqrt [4]{-1} a\\ r(t)&\to \sqrt [4]{-1} a\\ r(t)&\to -(-1)^{3/4} a\\ r(t)&\to (-1)^{3/4} a \end{align*}

✓ Sympy. Time used: 0.698 (sec). Leaf size: 32

from sympy import * 
t = symbols("t") 
a = symbols("a") 
r = Function("r") 
ode = Eq(-sqrt(a**8 - r(t)**8) + r(t)**3*Derivative(r(t), t),0) 
ics = {} 
dsolve(ode,func=r(t),ics=ics)
 

\[ \int \limits ^{r{\left (t \right )}} \frac {y^{3}}{\sqrt {\left (- y + a\right ) \left (y^{2} + a^{2}\right ) \left (y^{4} + a^{4}\right ) \left (y + a\right )}}\, dy = C_{1} + t \]