15.1.19 problem 19

Internal problem ID [2859]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 5, page 21
Problem number : 19
Date solved : Sunday, March 30, 2025 at 12:34:42 AM
CAS classification : [_separable]

\begin{align*} \tan \left (x \right ) \sin \left (x \right )^{2}+\cos \left (x \right )^{2} \cot \left (y\right ) y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.059 (sec). Leaf size: 40

ode:=tan(x)*sin(x)^2+cos(x)^2*cot(y(x))*diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \arcsin \left (\frac {\sqrt {2}\, \sqrt {\frac {1}{\cos \left (2 x \right )+1}}\, {\mathrm e}^{\frac {-1+\cos \left (2 x \right )}{2 \cos \left (2 x \right )+2}}}{c_1}\right ) \]

✓ Mathematica. Time used: 12.583 (sec). Leaf size: 24

ode=Tan[x]*Sin[x]^2+Cos[x]^2*Cot[y[x]]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \arcsin \left (\frac {1}{8} c_1 e^{-\frac {1}{2} \sec ^2(x)} \sec (x)\right ) \]

✓ Sympy. Time used: 1.651 (sec). Leaf size: 39

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

\[ \left [ y{\left (x \right )} = \pi - \operatorname {asin}{\left (\frac {C_{1} e^{- \frac {1}{2 \cos ^{2}{\left (x \right )}}}}{\cos {\left (x \right )}} \right )}, \ y{\left (x \right )} = \operatorname {asin}{\left (\frac {C_{1} e^{- \frac {1}{2 \cos ^{2}{\left (x \right )}}}}{\cos {\left (x \right )}} \right )}\right ] \]