83.3.8 problem 8

Internal problem ID [18985]
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 : 8
Date solved : Thursday, March 13, 2025 at 01:16:48 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.132 (sec). Leaf size: 40

ode:=1/x*cos(y(x))^2*diff(y(x),x)+1/y(x)*cos(x)^2 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ c_{1} +x^{2}+y \left (x \right )^{2}-1+\sin \left (2 x \right ) x +\frac {\cos \left (2 x \right )}{2}+y \left (x \right ) \sin \left (2 y \left (x \right )\right )+\frac {\cos \left (2 y \left (x \right )\right )}{2} = 0 \]

✓ Mathematica. Time used: 0.781 (sec). Leaf size: 65

ode=1/x*Cos[y[x]]^2*D[y[x],x]+( 1/y[x]*Cos[x]^2 )==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \text {InverseFunction}\left [2 \left (\frac {\text {$\#$1}^2}{4}+\frac {1}{4} \text {$\#$1} \sin (2 \text {$\#$1})+\frac {1}{8} \cos (2 \text {$\#$1})\right )\&\right ]\left [\frac {1}{4} \left (-\cos (2 x)-2 \left (x^2+x \sin (2 x)-2 c_1\right )\right )\right ] \]

✓ Sympy. Time used: 70.244 (sec). Leaf size: 88

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

\[ \frac {x^{2} \sin ^{2}{\left (x \right )}}{4} + \frac {x^{2} \cos ^{2}{\left (x \right )}}{4} + \frac {x \sin {\left (x \right )} \cos {\left (x \right )}}{2} + \frac {y^{2}{\left (x \right )} \sin ^{2}{\left (y{\left (x \right )} \right )}}{4} + \frac {y^{2}{\left (x \right )} \cos ^{2}{\left (y{\left (x \right )} \right )}}{4} + \frac {y{\left (x \right )} \sin {\left (y{\left (x \right )} \right )} \cos {\left (y{\left (x \right )} \right )}}{2} - \frac {\sin ^{2}{\left (y{\left (x \right )} \right )}}{4} + \frac {\cos ^{2}{\left (x \right )}}{4} = C_{1} \]