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) $y^{\prime }=(12x^2+5x)(y+9y^3),y(7)=11$

2) $y^{\prime }=ln(7x),y(-1)=10$

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

4) $y^{\prime }=xln(y-1),y(1)=1$

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

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

Separable equations

7) $y^{\prime }=y^{3}se{c}^2(x),$

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

9) $y^{\prime }=x^3/y^3$

10) $y^{\prime }=x^2+3x-9$

11) $y^{\prime }=cos(y)/sin(y)$

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

Initial value problems

13) $y^{\prime }=cos(x)+sin(x),y(0)=1$

14) $y^{\prime }=7y^2,y(5)=9$