Let's analyze and solve the given differential equations one by one. Here’s the breakdown of the equations:

10. $(1 + y^2) dx = (\tan^{-1} y - x) dy$

This is a differential equation. To solve it:

• Rearrange to separate variables or integrate directly if possible.

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

This equation is linear in $\frac{dy}{dx}$. Rewrite it for easier integration.

15. $\frac{dy}{dx} - y = e^{2x}$

This is a first-order linear differential equation. Solve using the integrating factor method:

• $\mu(x) = e^{\int -1 dx} = e^{-x}$

16. $\frac{dy}{dx} + y = -\frac{x}{y}$

This is a nonlinear equation. Rewrite as $y \frac{dy}{dx} + y^2 = -x$ and attempt to separate variables or apply substitution.

17. $2xy \frac{dy}{dx} = y^2 - x^2$

This equation can be separated into variables or solved by substitution $y = vx$.

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

This is an exact equation if $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, where $M(x, y)$ and $N(x, y)$ are the coefficients of $dx$ and $dy$, respectively.

Would you like me to solve these equations one at a time with step-by-step explanations? Let me know which one you'd like me to start with!