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