31.2.7 problem 7

Internal problem ID [5728]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 3
Problem number : 7
Date solved : Monday, January 27, 2025 at 01:11:40 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*}

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 43

dsolve((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,y(x), singsol=all)
 

\[ y \left (x \right ) = \frac {-n x +\operatorname {RootOf}\left (2 m^{2} x -2 n^{2} x -m \pi -2 \arcsin \left (\sin \left (\textit {\_Z} \right )+c_{1} \right ) m +2 \textit {\_Z} n \right )}{m} \]

✓ Solution by Mathematica

Time used: 0.656 (sec). Leaf size: 50

DSolve[(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,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}[\sin (m x) \sin (n y(x))-\cos (m x) \cos (n y(x))-\sin (n x) \cos (m y(x))-\cos (n x) \sin (m y(x))=c_1,y(x)] \]