We are tasked with solving the second-order non-homogeneous linear differential equation:

$y'' - 2y' + y = e^x$

Step 1: Solve the Homogeneous Equation

The first step is to solve the homogeneous equation:

$y'' - 2y' + y = 0$

To solve this, we find the characteristic equation associated with the differential equation:

$r^2 - 2r + 1 = 0$

Factoring the quadratic:

$(r - 1)^2 = 0$

Thus, we have a repeated root $r = 1$. Therefore, the general solution to the homogeneous equation is:

$y_h(x) = (C_1 + C_2 x) e^x$

where $C_1$ and $C_2$ are constants to be determined by initial conditions (if given).

Step 2: Find a Particular Solution

To find a particular solution to the non-homogeneous equation $y'' - 2y' + y = e^x$, we use the method of undetermined coefficients.

Since the right-hand side is $e^x$, which is part of the complementary solution $y_h(x)$, we multiply by $x$ to avoid duplication of the form already present in the homogeneous solution. Thus, we try a particular solution of the form:

$y_p(x) = A x^2 e^x$

where $A$ is a constant to be determined.

Now, compute the first and second derivatives of $y_p(x)$:

$y_p'(x) = 2A x e^x + A x^2 e^x$ $y_p''(x) = 2A e^x + 4A x e^x + A x^2 e^x$

Substitute these into the left-hand side of the differential equation $y'' - 2y' + y$:

$y_p'' - 2y_p' + y_p = (2A e^x + 4A x e^x + A x^2 e^x) - 2(2A x e^x + A x^2 e^x) + A x^2 e^x$

Simplifying:

$= 2A e^x + 4A x e^x + A x^2 e^x - 4A x e^x - 2A x^2 e^x + A x^2 e^x$ $= 2A e^x$

Now, set this equal to the right-hand side of the original equation, which is $e^x$:

$2A e^x = e^x$

Solve for $A$:

$2A = 1 \quad \Rightarrow \quad A = \frac{1}{2}$

Thus, the particular solution is:

$y_p(x) = \frac{1}{2} x^2 e^x$

Step 3: General Solution

The general solution to the differential equation is the sum of the homogeneous solution and the particular solution:

$y(x) = y_h(x) + y_p(x)$ $y(x) = (C_1 + C_2 x) e^x + \frac{1}{2} x^2 e^x$

Final Answer:

$y(x) = (C_1 + C_2 x) e^x + \frac{1}{2} x^2 e^x$

Do you need further explanation or details on any part of this solution? If you have initial conditions, I can help you find the specific values for $C_1$ and $C_2$.

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Here are some related questions you might explore:

1. How do you apply the method of undetermined coefficients to higher-order non-homogeneous differential equations?
2. What happens if the non-homogeneous term is not $e^x$, but a sine or cosine function?
3. How can you solve a non-homogeneous differential equation when the non-homogeneous term is a polynomial?
4. What is the distinction between a complementary solution and a particular solution in this context?
5. How do you handle cases where the non-homogeneous term is the same as part of the homogeneous solution?

Tip: Always be careful when choosing the form of the particular solution. If the non-homogeneous term matches any part of the complementary solution, multiply by an appropriate power of $x$ to ensure linear independence.