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To transform the fourth-order differential equation \( u^{(4)} - u = 0 \) into a system of first-order equations, let's break down the process:

### Step 1: Determine the number of new variables needed
Since the original equation is fourth-order, we need to introduce four new variables to reduce it to a system of first-order equations.

### Step 2: Define new variables
Let:
1. \( y_1 = u \)
2. \( y_2 = u' \)
3. \( y_3 = u'' \)
4. \( y_4 = u''' \)

This leads to the following relations:

\[
y_1' = y_2
\]
\[
y_2' = y_3
\]
\[
y_3' = y_4
\]
\[
y_4' = u^{(4)} = y_1
\]

### Step 3: Write the system of first-order equations
Using the definitions above, we can rewrite the original fourth-order equation as the following system:

\[
\begin{cases}
y_1' = y_2 \\
y_2' = y_3 \\
y_3' = y_4 \\
y_4' = y_1
\end{cases}
\]

Thus, the number of new variables needed is **4**, and the system of first-order equations is written as shown above.

Would you like further clarification on any step?

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Here are some related questions to expand your understanding:
1. How would you solve this system of first-order equations?
2. What is the general solution to this system?
3. How would the process change if the differential equation was of fifth order?
4. Can this system be represented in matrix form?
5. How do initial conditions affect the solutions of this system?

**Tip:** When transforming higher-order equations into a system of first-order equations, each derivative up to one order less than the original equation must be defined as a new variable.

Transform u^(4) - u = 0 into a system of first-order equations. First give the number of new variables needed, then write the equations.

Learn how to transform the fourth-order differential equation u^(4) - u = 0 into a system of first-order equations by introducing four new variables and writing the corresponding equations. This process is ideal for understanding differential systems at the college level.

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