Separable Ordinary Differential Equations:

Solve the following differential equations or IVP.

1. \( \displaystyle{(x+1)\frac{dy}{dx}=xy}\)   solution

2. \( \displaystyle{\frac{dy}{dx}+y^2\sin (x) = 0}\)   solution

3. \( \displaystyle{\frac{dy}{dx}= \cos^2 (x)\, \cos^2 (2y)} \)   solution

4. \(\displaystyle{x \frac{dy}{dx} = (1-y^2)^{1/2}} \)   solution

5. \( \displaystyle{x\frac{dy}{dx}=\cos(\ln x) }\)  solution

6. \( \displaystyle{(x^2-9) \frac{dy}{dx} +x y = 0} \)   solution

7. \( \displaystyle{(y^2+1)  dx = y \sec^2(x)  dy} \)   solution

8. \( \displaystyle{\frac{dy}{dx} = y^2-4} \)    solution

9. \( \displaystyle{x^2\frac{dy}{dx}=\sec^2(1/x) }\)   solution

10. \(\displaystyle{x\frac{dy}{dx}=1+y^2, \: y(-1)=1 }\)   solution

11. \( \displaystyle{\frac{dy}{dx}- (\sin x) y = 2 \sin x, \: y(\pi/2)=1}\)  solution   solution II