The solutions of `(dy)/(dx) =` |x| is

Options

A. `y = (x |x| )/(2) + C`
B. `y = (|x| )/(2) + C`
C. `y = (x^(2))/(2) + C`
D. `y = (x^(3))/(2) + C`

Answer

To solve the differential equation \(\frac{dy}{dx} = |x|\), we need to consider two cases based on the definition of the absolute value function.

### Step 1: Break down the absolute value function
The absolute value function \(|x|\) can be defined piecewise:
- When \(x \geq 0\), \(|x| = x\).
- When \(x < 0\), \(|x| = -x\).

### Step 2: Solve for \(x \geq 0\)
For the case when \(x \geq 0\):
\[ \frac{dy}{dx} = x \]
Integrate both sides:
\[ dy = x \, dx \]
Integrating gives:
\[ y = \frac{x^2}{2} + C_1 \]
where \(C_1\) is a constant of integration.

### Step 3: Solve for \(x < 0\)
For the case when \(x < 0\):
\[ \frac{dy}{dx} = -x \]
Integrate both sides:
\[ dy = -x \, dx \]
Integrating gives:
\[ y = -\frac{x^2}{2} + C_2 \]
where \(C_2\) is another constant of integration.

### Step 4: Combine the solutions
Now we can combine the two cases into a single expression. We can express the solution as:
\[ y = \frac{x^2}{2} + C \quad \text{for } x \geq 0 \]
\[ y = -\frac{x^2}{2} + C \quad \text{for } x < 0 \]
This can be compactly written using the absolute value:
\[ y = \frac{x^2}{2} + C \cdot \text{sgn}(x) \]
where \(\text{sgn}(x)\) is the sign function that returns \(1\) for \(x > 0\) and \(-1\) for \(x < 0\).

### Final Solution
Thus, the general solution can be expressed as:
\[ y = x \cdot |x| + C \]
where \(C\) is a constant.