60.1.76 problem 77

Internal problem ID [10090]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 77
Date solved : Sunday, March 30, 2025 at 03:14:28 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.029 (sec). Leaf size: 54

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

\[ y = \frac {-b x -2 \arctan \left (\frac {\tanh \left (\frac {\sqrt {a^{2}-b^{2}}\, \left (c_1 -x \right )}{2}\right ) \sqrt {a^{2}-b^{2}}}{a -b}\right )}{a} \]

✓ Mathematica. Time used: 0.498 (sec). Leaf size: 240

ode=D[y[x],x] - Cos[a*y[x]+b*x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}-\frac {\cos (b x+a K[2]) \int _1^x\left (\frac {a^3 \cos (b K[1]+a K[2]) \sin (b K[1]+a K[2])}{(b+a \cos (b K[1]+a K[2]))^2}-\frac {a^2 \sin (b K[1]+a K[2])}{b+a \cos (b K[1]+a K[2])}\right )dK[1] a+a+b \int _1^x\left (\frac {a^3 \cos (b K[1]+a K[2]) \sin (b K[1]+a K[2])}{(b+a \cos (b K[1]+a K[2]))^2}-\frac {a^2 \sin (b K[1]+a K[2])}{b+a \cos (b K[1]+a K[2])}\right )dK[1]}{b+a \cos (b x+a K[2])}dK[2]+\int _1^x\frac {a \cos (b K[1]+a y(x))}{b+a \cos (b K[1]+a y(x))}dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 56.528 (sec). Leaf size: 500

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

\[ \text {Solution too large to show} \]