Find a general solution to the differential equation.

y prime prime minus yy′′−yequals=minus−2222tplus+11

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Part 1

The general solution is

​y(t)equals=enter your response here.

​(Do not use​ d, D,​ e, E,​ i, or I as arbitrary constants since these letters already have defined​ meanings.)

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.

Find a general solution to the differential equation y'' - y = -2t + 1. (Do not use d, D, e, E, i, or I as arbitrary constants.)

Explore how to solve the second-order non-homogeneous differential equation y'' - y = -2t + 1 with a detailed step-by-step approach. The solution involves finding both the homogeneous and particular solutions using the method of undetermined coefficients. Suitable for advanced high school and college-level students.

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