The solution ` (dy)/(dx) = |x|` is :

Options

A. ` y = (x|x|)/2 +c`
B. ` y = (|x|)/2=c`
C. ` (dy)/(dx) = -x/y `
D. none of the above

Explanation

To solve the differential equation \(\frac{dy}{dx} = |x|\), we will consider two cases for the absolute value function.

### Step 1: Define the cases for \(|x|\)

The absolute value function \(|x|\) can be defined as:
- \(|x| = x\) when \(x \geq 0\)
- \(|x| = -x\) when \(x < 0\)

### Step 2: Solve for \(x \geq 0\)

For the case when \(x \geq 0\):
\[ \frac{dy}{dx} = x \]
Integrating both sides with respect to \(x\):
\[ y = \int x \, dx = \frac{x^2}{2} + C_1 \]

### Step 3: Solve for \(x < 0\)

For the case when \(x < 0\):
\[ \frac{dy}{dx} = -x \]
Integrating both sides with respect to \(x\):
\[ y = \int -x \, dx = -\frac{x^2}{2} + C_2 \]

### Step 4: Combine the results

Now we have two expressions for \(y\):
1. For \(x \geq 0\):
\[ y = \frac{x^2}{2} + C_1 \]
2. For \(x < 0\):
\[ y = -\frac{x^2}{2} + C_2 \]

We can combine these two results into a single expression using the absolute value:
\[ y = \frac{x^2}{2} + C \quad \text{for } x \geq 0 \]
\[ y = -\frac{x^2}{2} + C \quad \text{for } x < 0 \]

Thus, the complete solution can be expressed as:
\[ y = \frac{x \cdot |x|}{2} + C \]

### Final Answer
The solution to the differential equation \(\frac{dy}{dx} = |x|\) is:
\[ y = \frac{x \cdot |x|}{2} + C \]

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