60.1.79 problem 80

Internal problem ID [10093]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 80
Date solved : Sunday, March 30, 2025 at 03:15:29 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }+f \left (x \right ) \sin \left (y\right )+\left (1-f^{\prime }\left (x \right )\right ) \cos \left (y\right )-f^{\prime }\left (x \right )-1&=0 \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 41

ode:=diff(y(x),x)+f(x)*sin(y(x))+(1-diff(f(x),x))*cos(y(x))-diff(f(x),x)-1 = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = 2 \arctan \left (\frac {-{\mathrm e}^{\int f \left (x \right )d x}+\int {\mathrm e}^{\int f \left (x \right )d x}d x f \left (x \right )+c_1 f \left (x \right )}{c_1 +\int {\mathrm e}^{\int f \left (x \right )d x}d x}\right ) \]

✓ Mathematica. Time used: 7.206 (sec). Leaf size: 68

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

\begin{align*} y(x)\to 2 \arctan \left (f(x)+\frac {\exp \left (-\int _1^x-f(K[1])dK[1]\right )}{\int _1^x-\exp \left (-\int _1^{K[2]}-f(K[1])dK[1]\right )dK[2]+c_1}\right ) \\ y(x)\to 2 \arctan (f(x)) \\ \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE f(x)*sin(y(x)) - cos(y(x))*Derivative(f(x), x) + cos(y(x)) - Derivative(f(x), x) + Derivative(y(x), x) - 1 cannot be solved by the factorable group method