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Solve the ODE (CN) y ′′′ −3y ′′ +3y ′ −y=2e x x −2 ,x>0

User PEPEGA
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Answer:

To solve the given ordinary differential equation (ODE): y‴ - 3y″ + 3y′ - y = 2ex(x - 2), where x > 0, we can use the method of undetermined coefficients.

Homogeneous Solution:

First, let's find the homogeneous solution by assuming y = e^(rx), where r is a constant.

Substituting this into the ODE, we get the characteristic equation:

r³ - 3r² + 3r - 1 = 0

Factoring the equation, we have:

(r - 1)³ = 0

This gives us a triple root r = 1.

Therefore, the homogeneous solution is:

y_h = (C₁ + C₂x + C₃x²)e^x, where C₁, C₂, and C₃ are constants.

Particular Solution:

Next, we find a particular solution for the non-homogeneous term 2ex(x - 2). Since this term is a polynomial multiplied by an exponential function, we assume the particular solution has the form:

y_p = Ax²ex + Bxex

Now, we can calculate the derivatives of y_p:

y_p' = (2Ax + Bx²)ex + (Ax² + Bx)ex

y_p'' = (2A + 2Bx + 2Ax² + Bx²)ex + (2Ax + Bx²)ex

y_p''' = (2B + 6Ax + 6Bx² + 6Ax²)ex + (2A + 2Bx + 2Ax² + Bx²)ex

Substituting these derivatives into the ODE, we have:

(2B + 6Ax + 6Bx² + 6Ax²)ex + (2A + 2Bx + 2Ax² + Bx²)ex - 3[(2A + 2Bx + 2Ax² + Bx²)ex + (Ax² + Bx)ex] + (Ax² + Bx)ex - (Ax² + Bxex) = 2ex(x - 2)

Simplifying and grouping like terms, we get:

(6B - 3B - 2A + A)x²ex + (6A - 3A + B - B)xex + (2B + 2A)ex = 2ex(x - 2)

Comparing the coefficients of like terms, we have the following equations:

6B - 3B - 2A + A = 0 --> 3B - A = 0

6A - 3A + B - B = 0 --> 3A = 0

2B + 2A = 2 --> B + A = 1

From the second equation, we have A = 0.

Substituting this into the first and third equations, we get B = 0 and B + A = 1, respectively.

Therefore, we find that A = 0 and B = 1.

Hence, the particular solution is:

y_p = xex

General Solution:

The general solution is the sum of the homogeneous and particular solutions:

y = y_h + y_p = (C₁ + C₂x + C₃x²)e^x + xex

So, the general solution to the given ODE is:

y = (C₁ + C₂x + C₃x²)e^x + xex, where C₁, C₂, and C₃ are arbitrary constants.

Explanation:

User Malyy
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