Solve dy/dx=(3x+y+4)2

Question

solve this by variable separable method

Answer 1

Assuming you mean dy/dx = (3x + y + 4)^2

Let v = 3x + y + 4

giving, dy/dx = v^2, and dv/dx = 3 + dy/dx

Therefore, dv/dx = 3 + v^2

rearranging as,   dv/(3+v^2) = dx  (here is where the variables are separated)

or,   dv/(1 + (v/rt(3))^2) = 3dx       (to be on their own side of the equal sign)

integrating both sides,

rt(3).tan^(-1)(v/rt(3)) = 3x + C

tan^(-1)((3x+y+4)/rt(3)) = 3.rt(3)x + C2

3x + y + 4 = rt(3).tan(3.rt(3).x + C2)