77.5.21 problem 21

Internal problem ID [20401]
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 (D) at page 16
Problem number : 21
Date solved : Thursday, October 02, 2025 at 05:52:41 PM
CAS classification : [_Bernoulli]

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

✓ Maple. Time used: 0.088 (sec). Leaf size: 49

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

\[ y = \left (\frac {{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 n -{\mathrm e}^{\sin \left (x \right ) \left (n -1\right )} c_1 +2 \sin \left (x \right ) n -2 \sin \left (x \right )+2}{n -1}\right )^{-\frac {1}{n -1}} \]

✓ Mathematica. Time used: 5.901 (sec). Leaf size: 36

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

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

✓ Sympy. Time used: 4.416 (sec). Leaf size: 228

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

\[ y{\left (x \right )} = \begin {cases} \left (\frac {C_{1} n^{2} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} - \frac {2 C_{1} n e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {C_{1} e^{n \sin {\left (x \right )}}}{n^{2} e^{\sin {\left (x \right )}} - 2 n e^{\sin {\left (x \right )}} + e^{\sin {\left (x \right )}}} + \frac {2 n^{2} \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {4 n \sin {\left (x \right )}}{n^{2} - 2 n + 1} + \frac {2 n}{n^{2} - 2 n + 1} + \frac {2 \sin {\left (x \right )}}{n^{2} - 2 n + 1} - \frac {2}{n^{2} - 2 n + 1}\right )^{- \frac {1}{n - 1}} & \text {for}\: n > 1 \vee n < 1 \\\left (C_{1} e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} + n e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \cos ^{2}{\left (x \right )} - e^{n \sin {\left (x \right )} - \sin {\left (x \right )}} \cos ^{2}{\left (x \right )}\right )^{- \frac {1}{n - 1}} & \text {otherwise} \end {cases} \]