9.7.9 problem 9

Internal problem ID [2990]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 11, page 45
Problem number : 9
Date solved : Tuesday, September 30, 2025 at 06:14:04 AM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \csc \left (y\right ) \cot \left (y\right ) y^{\prime }&=\csc \left (y\right )+{\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.006 (sec). Leaf size: 16

ode:=csc(y(x))*cot(y(x))*diff(y(x),x) = csc(y(x))+exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {arccsc}\left ({\mathrm e}^{-x} c_1 -\frac {{\mathrm e}^{x}}{2}\right ) \]

✓ Mathematica. Time used: 0.631 (sec). Leaf size: 30

ode=Csc[y[x]]*Cot[y[x]]*D[y[x],x]==(Csc[y[x]]+Exp[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\csc ^{-1}\left (\frac {e^x}{2}-c_1 e^{-x}\right )\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.849 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-exp(x) - 1/sin(y(x)) + Derivative(y(x), x)/(sin(y(x))*tan(y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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