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The uploaded image contains three differential equations that need to be solved. Here's a breakdown:

### Equation 16:
\[
(D^2 - 4D + 4)y = e^{2x} + x^3 + \cos(2x)
\]
Where \( D \) represents differentiation with respect to \( x \). We need to solve this second-order linear differential equation with the given non-homogeneous terms.

### Equation 17:
\[
(D^2 - 3D + 2)y = x + \sin(x)
\]
This is also a second-order linear differential equation with non-homogeneous terms \( x \) and \( \sin(x) \).

### Equation 18:
\[
(D^5 - D)y = 12e^x + 85mx + 2
\]
This is a fifth-order differential equation with a non-homogeneous right-hand side involving an exponential function and a polynomial.

To solve these equations, the general procedure would involve:
1. Solving the **homogeneous equation**.
2. Finding a **particular solution** for the non-homogeneous part using methods such as undetermined coefficients or variation of parameters.
3. Combining both to find the general solution.

Would you like me to go through each equation one by one, or do you have a specific one you'd like to focus on first?

Solve the differential equations: (D^2 - 4D + 4)y = e^(2x) + x^3 + cos(2x), (D^2 - 3D + 2)y = x + sin(x), (D^5 - D)y = 12e^x + 85mx + 2.

Explore solutions to second and fifth order non-homogeneous linear differential equations. This detailed guide covers solving the characteristic equation for the homogeneous part, using undetermined coefficients to find particular solutions, and applying the superposition principle.

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