9.8.30 problem 31

Internal problem ID [3033]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 12, page 46
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 06:23:02 AM
CAS classification : [_separable]

\begin{align*} 3 \,{\mathrm e}^{x} \tan \left (y\right )&=\left (1-{\mathrm e}^{x}\right ) \sec \left (y\right )^{2} y^{\prime } \end{align*}

✓ Maple. Time used: 0.026 (sec). Leaf size: 134

ode:=3*exp(x)*tan(y(x)) = (1-exp(x))*sec(y(x))^2*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\arctan \left (-\frac {2 c_1 \left (-1+{\mathrm e}^{x}\right )^{3}}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_1^{2}-6 \,{\mathrm e}^{x}+1}, \frac {-c_1^{2}+15 \,{\mathrm e}^{2 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{4 x}-6 \,{\mathrm e}^{5 x}+{\mathrm e}^{6 x}-6 \,{\mathrm e}^{x}+1}{{\mathrm e}^{6 x}-6 \,{\mathrm e}^{5 x}+15 \,{\mathrm e}^{4 x}-20 \,{\mathrm e}^{3 x}+15 \,{\mathrm e}^{2 x}+c_1^{2}-6 \,{\mathrm e}^{x}+1}\right )}{2} \]

✓ Mathematica. Time used: 0.741 (sec). Leaf size: 78

ode=3*Exp[x]*Tan[y[x]]==(1-Exp[x])*Sec[y[x]]^2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\right )+2 c_1\right )\right )\\ y(x)&\to \frac {1}{2} \arccos \left (-\tanh \left (-3 \log \left (2-2 e^x\right )+2 c_1\right )\right )\\ y(x)&\to 0\\ y(x)&\to -\frac {\pi }{2}\\ y(x)&\to \frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 1.807 (sec). Leaf size: 175

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((exp(x) - 1)*Derivative(y(x), x)/cos(y(x))**2 + 3*exp(x)*tan(y(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \pi - \frac {\operatorname {acos}{\left (\frac {- C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1}{C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1} \right )}}{2}, \ y{\left (x \right )} = \frac {\operatorname {acos}{\left (\frac {- C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1}{C_{1} - e^{6 x} + 6 e^{5 x} - 15 e^{4 x} + 20 e^{3 x} - 15 e^{2 x} + 6 e^{x} - 1} \right )}}{2}\right ] \]