Hello please solve it fast

Finde general solutions for the following differential equations

(D²-2D+1)y=2x+2

To find the general solution for the differential equation \((D^2 - 2D + 1)y = 2x + 2\), we need to break it down into two parts: the homogeneous solution and the particular solution. Let’s tackle this step by step.

Step 1: Homogeneous Solution

The first step is to solve the associated homogeneous equation, which is given by:

(D² - 2D + 1)y = 0

This can be factored as:

(D - 1)²y = 0

To find the characteristic equation, we set:

r² - 2r + 1 = 0

This simplifies to:

(r - 1)² = 0

Thus, we have a repeated root at \(r = 1\). The general solution for the homogeneous part is given by:

y_h = C₁e^x + C₂xe^x

where \(C₁\) and \(C₂\) are constants determined by initial conditions.

Step 2: Particular Solution

Next, we need to find a particular solution \(y_p\) for the non-homogeneous equation:

(D² - 2D + 1)y = 2x + 2

Since the right-hand side is a polynomial of degree 1, we can assume a particular solution of the form:

y_p = Ax + B

where \(A\) and \(B\) are constants to be determined. Now, we need to compute the derivatives:

• D(y_p) = A
• D²(y_p) = 0

Substituting these into the left-hand side of the differential equation gives:

(0 - 2A + 1(Ax + B)) = 2x + 2

This simplifies to:

(A - 2A + B) = 2x + 2

or:

-A + B = 2

Now, we can equate coefficients. For the \(x\) terms, we have:

-A = 2, which gives A = -2.

For the constant terms, we have:

B - 2 = 2, which gives B = 4.

Thus, the particular solution is:

y_p = -2x + 4.

Step 3: General Solution

Now that we have both the homogeneous and particular solutions, we can combine them to form the general solution:

y = y_h + y_p

Substituting in our solutions, we get:

y = C₁e^x + C₂xe^x - 2x + 4.

Final Result

In conclusion, the general solution to the differential equation \((D^2 - 2D + 1)y = 2x + 2\) is:

y = C₁e^x + C₂xe^x - 2x + 4.

This solution encompasses both the behavior dictated by the homogeneous part and the specific response to the non-homogeneous input. If you have any further questions or need clarification on any part of the process, feel free to ask!