4.8.24 \(x^4 y'(x)+x^3 y(x)+\csc (x y(x))=0\)

ODE
\[ x^4 y'(x)+x^3 y(x)+\csc (x y(x))=0 \] ODE Classification

[[_1st_order, `_with_symmetry_[F(x),G(x)*y+H(x)]`]]

Book solution method
Change of Variable, new dependent variable

Mathematica
cpu = 1.51103 (sec), leaf count = 40

\[\left \{\left \{y(x)\to -\frac {\cos ^{-1}\left (c_1-\frac {1}{2 x^2}\right )}{x}\right \},\left \{y(x)\to \frac {\cos ^{-1}\left (c_1-\frac {1}{2 x^2}\right )}{x}\right \}\right \}\]

Maple
cpu = 0.405 (sec), leaf count = 23

\[ \left \{ -{\it \_C1}+{\frac {-2\,\cos \left (xy \relax (x ) \right ) {x}^{2}-1}{{x}^{2}}}=0 \right \} \] Mathematica raw input

DSolve[Csc[x*y[x]] + x^3*y[x] + x^4*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(ArcCos[-1/(2*x^2) + C[1]]/x)}, {y[x] -> ArcCos[-1/(2*x^2) + C[1]]/x}
}

Maple raw input

dsolve(x^4*diff(y(x),x)+x^3*y(x)+csc(x*y(x)) = 0, y(x),'implicit')

Maple raw output

-_C1+(-2*cos(x*y(x))*x^2-1)/x^2 = 0