Ordinary Differential Equations/Separable 3

Existence problems
Tell whether the following initial value problems have a solution or not, and if its solution is unique.

1) $$y'=(12x^2+5x)(y+9y^3),y(7)=11$$

2) $$y'=ln(7x),y(-1)=10$$

3) $$y'=\frac{x+7x^2-6x^3}{y^2-1},y(0)=16$$

4) $$y'=xln(y-1),y(1)=1$$

5) $$y'=\frac{x^3+5x}{y^2+7y+12},y(5)=9$$

6) $$y'=\frac{y+7y^2}{x-5},y(5)=4$$

Separable equations
7) $$y'=y^3sec^2(x),$$

8) $$y'=\frac{5y^2+6}{y}$$

9) $$y'=x^3/y^3$$

10) $$y'=x^2+3x-9$$

11) $$y'=cos(y)/sin(y)$$

12) $$y'=\frac{cos(x)}{sin(y)}$$

Initial value problems
13) $$y'=cos(x)+sin(x),y(0)=1$$

14) $$y'=7y^2,y(5)=9$$