83.19.8 problem 8

Internal problem ID [19165]
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 : 8
Date solved : Monday, March 31, 2025 at 06:50:48 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.021 (sec). Leaf size: 32

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

\begin{align*} y &= 0 \\ x -\int _{}^{y}\frac {1}{\operatorname {RootOf}\left (\textit {\_a} -\sin \left (\textit {\_Z} \right )+\textit {\_Z} \cos \left (\textit {\_Z} \right )\right )}d \textit {\_a} -c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 5.193 (sec). Leaf size: 174

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

\begin{align*} y(x)\to (x-c_1) \arccos (-x+c_1)-\sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to (-x+c_1) \arccos (-x+c_1)-\sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to (x-c_1) \arccos (-x+c_1)+\sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to (-x+c_1) \arccos (-x+c_1)+\sqrt {-x^2+2 c_1 x+1-c_1{}^2} \\ y(x)\to 0 \\ \end{align*}

✗ Sympy

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

NotImplementedError : multiple generators [_X0, tan(_X0/2)] 
No algorithms are implemented to solve equation _X0*(1 - tan(_X0/2)**2)/(tan(_X0/2)**2 + 1) + y(x) - 2*tan(_X0/2)/(tan(_X0/2)**2 + 1)