Separate the variables:
dy/dx=x^3y^4; y^-4dy=x^3dx; ∫y^-4=∫x^3dx; -y^-3/3=x^4/4+C; -4y^-3=3x^4+12C;
-4=3x^4y^3+12Cy^3; 4=-3x^4y^3-12Cy^3; 4=y^3(-3x^4-12C); 4=y^3(K-3x^4), where K is a constant (derived from constant C).
CHECK
Differentiate: 0=y^3(-12x^3)+3(K-3x^4)y^2y'; 0=-4yx^3+(K-3x^4)y'; 4yx^3=4y'/y^3; x^3y^4=y'. Confirmed.