Find a general solution to the differential equation y''-y=-22t+2

We are given the second-order linear differential equation:

\[
y'' - y = -22t + 2
\]

To solve this, we can break the solution into two parts:

1. **Homogeneous solution**: Solve the homogeneous equation \( y'' - y = 0 \).
2. **Particular solution**: Find a particular solution to the non-homogeneous equation \( y'' - y = -22t + 2 \).

### Step 1: Solve the homogeneous equation

The homogeneous equation is:

\[
y'' - y = 0
\]

This can be rewritten as:

\[
y'' = y
\]

The characteristic equation for this is:

\[
r^2 - 1 = 0
\]

Factoring this gives:

\[
(r - 1)(r + 1) = 0
\]

Thus, \( r = 1 \) and \( r = -1 \) are the roots. The general solution to the homogeneous equation is:

\[
y_h(t) = c_1 e^t + c_2 e^{-t}
\]

where \( c_1 \) and \( c_2 \) are constants to be determined later.

### Step 2: Find a particular solution

Now, we need to find a particular solution to the non-homogeneous equation:

\[
y'' - y = -22t + 2
\]

We use the method of undetermined coefficients. Since the right-hand side is a linear polynomial, we try a particular solution of the form:

\[
y_p(t) = At + B
\]

where \( A \) and \( B \) are constants to be determined.

First, we find the first and second derivatives of \( y_p(t) \):

\[
y_p'(t) = A
\]
\[
y_p''(t) = 0
\]

Substitute these into the original equation \( y'' - y = -22t + 2 \):

\[
0 - (At + B) = -22t + 2
\]

This simplifies to:

\[
-At - B = -22t + 2
\]

Now, equate the coefficients of like powers of \( t \) on both sides:

- For the coefficient of \( t \): \( -A = -22 \), so \( A = 22 \).
- For the constant term: \( -B = 2 \), so \( B = -2 \).

Thus, the particular solution is:

\[
y_p(t) = 22t - 2
\]

### Step 3: General solution

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

\[
y(t) = y_h(t) + y_p(t)
\]

Substitute the expressions for \( y_h(t) \) and \( y_p(t) \):

\[
y(t) = c_1 e^t + c_2 e^{-t} + 22t - 2
\]

This is the general solution to the differential equation.

### Summary

The general solution to the differential equation \( y'' - y = -22t + 2 \) is:

\[
y(t) = c_1 e^t + c_2 e^{-t} + 22t - 2
\]

Would you like more details on a particular part of the solution?

### 5 Related Questions:
1. How would you solve the differential equation \( y'' - y = t^2 + 3t \)?
2. What is the method of undetermined coefficients, and when is it used?
3. Can this equation be solved using variation of parameters?
4. How do you determine the characteristic equation for higher-order differential equations?
5. How do different forms of forcing functions affect the choice of particular solutions?

### Tip:
Always check the form of the non-homogeneous term before deciding on a method for finding the particular solution. This helps in selecting the correct form for the undetermined coefficients.

Find a general solution to the differential equation y'' - y = -22t + 2.

Discover how to solve the second-order differential equation y'' - y = -22t + 2 step by step. This solution includes finding the homogeneous and particular solutions using the method of undetermined coefficients. Suitable for students in undergraduate calculus or differential equations courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation