60.1.501 problem 514

Internal problem ID [10515]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 514
Date solved : Sunday, March 30, 2025 at 05:42:03 PM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2} \left (a \cos \left (y\right )+b \right )-c \cos \left (y\right )+d&=0 \end{align*}

✓ Maple. Time used: 0.011 (sec). Leaf size: 84

ode:=diff(y(x),x)^2*(a*cos(y(x))+b)-c*cos(y(x))+d = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \arccos \left (\frac {d}{c}\right ) \\ x -\int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_1 &= 0 \\ x +\int _{}^{y}\frac {a \cos \left (\textit {\_a} \right )+b}{\sqrt {\left (a \cos \left (\textit {\_a} \right )+b \right ) \left (c \cos \left (\textit {\_a} \right )-d \right )}}d \textit {\_a} -c_1 &= 0 \\ \end{align*}

✓ Mathematica. Time used: 11.716 (sec). Leaf size: 627

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Sympy. Time used: 62.045 (sec). Leaf size: 48

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

\[ \left [ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {c \cos {\left (y \right )} - d}{a \cos {\left (y \right )} + b}}}\, dy = C_{1} - x, \ \int \limits ^{y{\left (x \right )}} \frac {1}{\sqrt {\frac {c \cos {\left (y \right )} - d}{a \cos {\left (y \right )} + b}}}\, dy = C_{1} + x\right ] \]