27.2.3 problem 3

Internal problem ID [4303]
Book : An introduction to the solution and applications of differential equations, J.W. Searl, 1966
Section : Chapter 4, Ex. 4.2
Problem number : 3
Date solved : Tuesday, March 04, 2025 at 06:19:23 PM
CAS classification : [_separable]

\begin{align*} r y^{\prime }&=\frac {\left (a^{2}-r^{2}\right ) \tan \left (y\right )}{a^{2}+r^{2}} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 17

ode:=r*diff(y(r),r) = (a^2-r^2)/(a^2+r^2)*tan(y(r)); 
dsolve(ode,y(r), singsol=all);
 

\[ y \left (r \right ) = \arcsin \left (\frac {r c_{1}}{a^{2}+r^{2}}\right ) \]

✓ Mathematica. Time used: 23.758 (sec). Leaf size: 26

ode=r*D[y[r],r]== (a^2-r^2)/(a^2+r^2)*Tan[y[r]]; 
ic={}; 
DSolve[{ode,ic},y[r],r,IncludeSingularSolutions->True]
 

\begin{align*} y(r)\to \arcsin \left (\frac {e^{c_1} r}{a^2+r^2}\right ) \\ y(r)\to 0 \\ \end{align*}

✓ Sympy. Time used: 0.534 (sec). Leaf size: 29

from sympy import * 
r = symbols("r") 
a = symbols("a") 
y = Function("y") 
ode = Eq(r*Derivative(y(r), r) - (a**2 - r**2)*tan(y(r))/(a**2 + r**2),0) 
ics = {} 
dsolve(ode,func=y(r),ics=ics)
 

\[ \left [ y{\left (r \right )} = \pi - \operatorname {asin}{\left (\frac {C_{1} r}{a^{2} + r^{2}} \right )}, \ y{\left (r \right )} = \operatorname {asin}{\left (\frac {C_{1} r}{a^{2} + r^{2}} \right )}\right ] \]