https://file.aiquickdraw.com/mathbot/file/172881645499008xw72t2.jpg

The image contains six differential equations, and you are asked to solve any three of them. Here's a brief overview of the equations provided:

(a) \( e^x(y^2 + 1) \, dx = 2y(e^x + 1) \, dy \)

(b) \( \frac{dy}{dx} = \frac{7x - 3y - 7}{7y - 3x + 3} \)

(c) \( (x^2 + y^2) \, dx - 2xy \, dy = 0 \)

(d) \( (x^2 - 4xy - 2y^2) \, dx + (y^2 - 4xy - 2x^2) \, dy = 0 \)

(e) \( \frac{d^2y}{dx^2} - \frac{2xy}{x^2 + 1} = \frac{4x^2}{x^2 + 1} \)

(f) \( \frac{d^2y}{dx^2} - 5 \frac{dy}{dx} + 6y = e^{4x} \)

Would you like to solve all three or focus on specific ones? Let me know if you need detailed steps for any equation.

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Here are some possible next steps you might want to explore:
1. Solve equation (a) in detail using a method like separation of variables.
2. Attempt equation (b) and identify whether it’s a homogeneous equation.
3. Solve equation (f) using the method of undetermined coefficients.
4. Explore exactness for equation (d) to check if it's an exact differential equation.
5. Investigate the characteristics of the system in equation (e) by rewriting it in a simpler form.

**Tip:** Identifying the type of differential equation (e.g., homogeneous, exact, linear) helps in choosing the most efficient solution method.

Solve any three of the following differential equations:
(a) e^x(y^2 + 1) dx = 2y(e^x + 1) dy
(b) dy/dx = (7x - 3y - 7) / (7y - 3x + 3)
(c) (x^2 + y^2) dx - 2xy dy = 0
(d) (x^2 - 4xy - 2y^2) dx + (y^2 - 4xy - 2x^2) dy = 0
(e) d^2y/dx^2 - (2xy) / (x^2 + 1) = 4x^2 / (x^2 + 1)
(f) d^2y/dx^2 - 5(dy/dx) + 6y = e^(4x)

Explore the solution techniques for differential equations, including separation of variables, linear equations, and exact equations. This page covers example problems such as e^x(y^2 + 1) dx = 2y(e^x + 1) dy, and second-order linear equations with non-homogeneous terms. Ideal for undergraduate math and engineering students.

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