83.19.9 problem 9

Internal problem ID [19166]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (B) at page 55
Problem number : 9
Date solved : Monday, March 31, 2025 at 06:50:51 PM
CAS classification : [_linear]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 34

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

\[ y = \frac {2 \cot \left (x \right )-\pi +2 \csc \left (x \right )+2 x +2 c_1}{2 \csc \left (x \right )+2 \cot \left (x \right )} \]

✓ Mathematica. Time used: 0.159 (sec). Leaf size: 44

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

\[ y(x)\to \frac {1}{2} e^{-\text {arctanh}(\cos (x))} \left (\sqrt {\sin ^2(x)} \left (\csc ^2\left (\frac {x}{2}\right )+2 x \csc (x)\right )+2 c_1\right ) \]

✓ Sympy. Time used: 142.414 (sec). Leaf size: 99

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

\[ y{\left (x \right )} = \frac {\sqrt {\cos {\left (x \right )} - 1} \left (C_{1} - \int \frac {\sqrt {\cos {\left (x \right )} + 1}}{\sqrt {\cos {\left (x \right )} - 1} \tan {\left (x \right )}}\, dx + \int \frac {\sqrt {\cos {\left (x \right )} + 1} y{\left (x \right )}}{\sqrt {\cos {\left (x \right )} - 1} \sin {\left (x \right )}}\, dx\right )}{\sqrt {\cos {\left (x \right )} - 1} \int \frac {\sqrt {\cos {\left (x \right )} + 1}}{\sqrt {\cos {\left (x \right )} - 1} \sin {\left (x \right )}}\, dx + \sqrt {\cos {\left (x \right )} + 1}} \]