25.2.7 problem 7

Internal problem ID [6863]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 3
Problem number : 7
Date solved : Tuesday, September 30, 2025 at 03:56:08 PM
CAS classification : [_exact]

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

✓ Maple. Time used: 0.015 (sec). Leaf size: 36

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

\[ y = \frac {-m x +\operatorname {RootOf}\left (-m^{2} x +n^{2} x +\arcsin \left (\cos \left (\textit {\_Z} \right )+c_1 \right ) n +\textit {\_Z} m \right )}{n} \]

✓ Mathematica. Time used: 0.349 (sec). Leaf size: 108

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

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

✗ Sympy

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

Timed Out