Here is a proper formatting of the differential equations provided in the uploaded image:

Solve the following differential equations:

1. $(x^2 + 1)\frac{dy}{dx} + 4xy = \frac{1}{x^2 + 1}$

2. $(3x^2y^4 + 4xy)dx + (2x^3y^3 - x^2)dy = 0$

3. $\frac{dy}{dx} + x \sin^2 y = x^2 \cos^2 y$

4. $(1 - x^2)\frac{dy}{dx} + 2xy = x \sqrt{1 - x^2}$

5. $(xy^2 \sin y + x \cos y)dx + (x^2 y \sin y - x \cos y)dy = 0$

6. $(2x^3 + xy^3)dy + (x^2y + y^4)dx = 0$

7. $(xy^2 - e^{x^3})dx - x^2dy = 0$

8. $(D^4 + xD^2 + 1)y = x^2 \cos x$

9. $(3x + 2)^2\frac{d^2y}{dx^2} + (8x + 6)\frac{dy}{dx} - 36y = 3x^2 + 4x + 1$

10. $\frac{d^2y}{dx^2} - y = e^{-x} \sin(e^{-x}) + \cos(e^{-x}) \quad [\text{MOVOP}]$

11. $\frac{d^2y}{dx^2} + 3\frac{dy}{dx} + 2y = e^x$

12. $\frac{d^2y}{dx^2} + 4y = \tan 2x \quad [\text{MOVOP}]$

13. $x^2 \frac{d^2y}{dx^2} + x \frac{dy}{dx} + y = \log x \cdot \sin(\log x)$

14. $\frac{dx}{dt} + 2y = e^t, \quad \frac{dy}{dt} - 2x = e^t \quad [\text{Simultaneous}]$

Would you like a solution to any particular question?