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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.

Solve the following differential equations: 
1. dy/dx = 3x^2 - 6x + 5 
2. x(dy/dx) = 5x^3 + 4 
3. e^x (dy/dx) = 4 
4. dy/dx = 2x / (y + 1) 
5. dy/dx = (1 + x)(1 + y) 
6. dy/dx = (1 + y) / (2x) 
7. dy/dx = (y^2 + xy^2) / (x^2 - y^2) 
8. dy/dx = (y^2 - 1) / x 
9. x(dy/dx) = (x^2 + 1) / (y + 1) 
10. x(dy/dx) = y + xy 
11. y + x (dy/dx) = (4 + y^2) sec^2 x 
12. sin(x)(dy/dx) = cos(x) / (1 + y) 
13. dy/dx = (x + 5y) / 2x 
14. dy/dx = (x^2 + y^2) / xy

Learn how to solve a set of first-order and separable differential equations, including key topics like separation of variables, exponential equations, and integrals. Suitable for undergraduate students studying differential equations.

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