77.3.12 problem 12

Internal problem ID [20353]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Exercise II (B) at page 9
Problem number : 12
Date solved : Thursday, October 02, 2025 at 05:45:37 PM
CAS classification : [_separable]

\begin{align*} \cos \left (y\right ) \ln \left (\sec \left (x \right )+\tan \left (x \right )\right )&=\cos \left (x \right ) \ln \left (\sec \left (y\right )+\tan \left (y\right )\right ) y^{\prime } \end{align*}

✓ Maple. Time used: 0.222 (sec). Leaf size: 91

ode:=cos(y(x))*ln(sec(x)+tan(x)) = cos(x)*ln(sec(y(x))+tan(y(x)))*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \arctan \left (\tanh \left (\sqrt {\ln \left (-\frac {\cos \left (x \right )}{\sin \left (x \right )-1}\right )^{2}+2 c_1}\right ), \operatorname {sech}\left (\sqrt {\ln \left (-\frac {\cos \left (x \right )}{\sin \left (x \right )-1}\right )^{2}+2 c_1}\right )\right ) \\ y &= \arctan \left (-\tanh \left (\sqrt {\ln \left (-\frac {\cos \left (x \right )}{\sin \left (x \right )-1}\right )^{2}+2 c_1}\right ), \operatorname {sech}\left (\sqrt {\ln \left (-\frac {\cos \left (x \right )}{\sin \left (x \right )-1}\right )^{2}+2 c_1}\right )\right ) \\ \end{align*}

✓ Mathematica. Time used: 60.156 (sec). Leaf size: 103

ode=Cos[y[x]]*Log[ Sec[x]+Tan[x] ]==Cos[x]*Log[Sec[y[x]]+Tan[y[x]]]*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\sec ^{-1}\left (\frac {1}{2} \left (e^{-\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}+e^{\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}\right )\right )\\ y(x)&\to \sec ^{-1}\left (\frac {1}{2} \left (e^{-\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}+e^{\sqrt {\log ^2(\tan (x)+\sec (x))+2 c_1}}\right )\right ) \end{align*}

✓ Sympy. Time used: 43.081 (sec). Leaf size: 32

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

\[ \int \limits ^{y{\left (x \right )}} \frac {\log {\left (\tan {\left (y \right )} + \frac {1}{\cos {\left (y \right )}} \right )}}{\cos {\left (y \right )}}\, dy = C_{1} + \int \frac {\log {\left (\tan {\left (x \right )} + \frac {1}{\cos {\left (x \right )}} \right )}}{\cos {\left (x \right )}}\, dx \]