Let's go through the differential equations one by one. I'll solve the first few equations, then let me know if you'd like further details or assistance with the remaining ones.

(1) $\frac{dy}{dx} = 3x^2 - 6x + 5$

This is a simple first-order differential equation. We can solve it by integrating both sides with respect to $x$.

$y = \int (3x^2 - 6x + 5) \, dx$

The integral is:

$y = x^3 - 3x^2 + 5x + C$

where $C$ is the constant of integration.

(2) $x \frac{dy}{dx} = 5x^3 + 4$

This is separable. First, divide both sides by $x$:

$\frac{dy}{dx} = 5x^2 + \frac{4}{x}$

Now, integrate both sides:

$y = \int \left( 5x^2 + \frac{4}{x} \right) \, dx$

The result is:

$y = \frac{5}{3}x^3 + 4\ln|x| + C$

where $C$ is the constant of integration.

(3) $e^x \frac{dy}{dx} = 4$

First, divide by $e^x$:

$\frac{dy}{dx} = 4e^{-x}$

Now integrate both sides with respect to $x$:

$y = \int 4e^{-x} \, dx$

The integral of $e^{-x}$ is $-e^{-x}$, so:

$y = -4e^{-x} + C$

(4) $\frac{dy}{dx} = \frac{2x}{y+1}$

This is a separable differential equation. Rearrange to separate variables:

$(y+1) dy = 2x dx$

Now, integrate both sides:

$\int (y+1) \, dy = \int 2x \, dx$

$\frac{y^2}{2} + y = x^2 + C$

This is the implicit solution to the equation.

(5) $\frac{dy}{dx} = (1 + x)(1 + y)$

This is also a separable equation. We can rewrite it as:

$\frac{1}{1 + y} \, dy = (1 + x) \, dx$

Integrate both sides:

$\int \frac{1}{1 + y} \, dy = \int (1 + x) \, dx$

$\ln|1 + y| = x + \frac{x^2}{2} + C$

Exponentiate both sides:

$1 + y = e^{x + \frac{x^2}{2} + C} = A e^{x + \frac{x^2}{2}}$

where $A = e^C$ is a constant. So the solution is:

$y = A e^{x + \frac{x^2}{2}} - 1$

Would you like me to continue with the remaining problems, or do you need further clarification on any of these? Here are some related problems you might consider:

1. How would you approach second-order differential equations?
2. What is the role of boundary conditions in solving differential equations?
3. Can a differential equation have more than one solution, and why?
4. What techniques are used to solve non-linear differential equations?
5. How do numerical methods (like Euler's method) help in solving complex differential equations?

Tip: For separable differential equations, always try to rearrange terms so that each variable (and its differential) is on one side of the equation before integrating.