The given differential equation is:

$y'' - y = -2t + 1$

This is a second-order non-homogeneous linear differential equation. To solve this, we follow the usual steps of solving such equations:

Step 1: Solve the Homogeneous Equation

First, solve the corresponding homogeneous equation:

$y'' - y = 0$

This can be rewritten as:

$y'' = y$

The characteristic equation for this is:

$r^2 - 1 = 0$

Solving for $r$:

$r = \pm 1$

So, the general solution to the homogeneous equation is:

$y_h(t) = c_1 e^t + c_2 e^{-t}$

Step 2: Solve the Non-Homogeneous Equation

Now, solve the non-homogeneous equation:

$y'' - y = -2t + 1$

We use the method of undetermined coefficients. Since the non-homogeneous term is a polynomial in $t$, we try a particular solution of the form:

$y_p(t) = At + B$

Now, find $y_p''(t)$:

$y_p'(t) = A, \quad y_p''(t) = 0$

Substitute into the original equation:

$0 - (At + B) = -2t + 1$

Simplify:

$-(At + B) = -2t + 1$

This gives:

$-At - B = -2t + 1$

Now, equate the coefficients of $t$ and the constant terms:

• For $t$: $-A = -2 \Rightarrow A = 2$
• For the constant: $-B = 1 \Rightarrow B = -1$

Thus, the particular solution is:

$y_p(t) = 2t - 1$

Step 3: General Solution

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

$y(t) = y_h(t) + y_p(t)$

So, the general solution is:

$y(t) = c_1 e^t + c_2 e^{-t} + 2t - 1$

Final Answer:

$y(t) = c_1 e^t + c_2 e^{-t} + 2t - 1$

Let me know if you would like further details or clarifications!

Related Questions:

1. How do you determine the form of the particular solution for different types of non-homogeneous terms?
2. What is the general process of solving second-order linear differential equations?
3. How do initial conditions influence the solution of a differential equation?
4. What are the characteristic equations, and how do they relate to solving homogeneous equations?
5. How do undetermined coefficients differ from variation of parameters?

Tip:

When solving non-homogeneous equations, always check whether the non-homogeneous term is similar to the solution of the homogeneous equation, as this may require modifying your guess for the particular solution.