What is the equation of the curve whose slope at any point is equal to 2x and which passes through the origin?

Options

A. A) `y(1-x)=x^(2)`
B. B) `y^(2)(1+x^(2)) =x^(4)`
C. C) `y^(2) =(x+1)`
D. D) `y=x^(2)`

Answer

To find the equation of the curve whose slope at any point is equal to \(2x\) and which passes through the origin, we can follow these steps:

### Step 1: Understand the relationship between slope and derivative
The slope of the curve at any point is given as \(2x\). In calculus, the slope of a curve at a point is represented by the derivative of the function, so we can write:
\[ \frac{dy}{dx} = 2x \]

### Step 2: Separate the variables
To solve this differential equation, we can separate the variables:
\[ dy = 2x \, dx \]

### Step 3: Integrate both sides
Now we will integrate both sides to find \(y\):
\[ \int dy = \int 2x \, dx \]
The left side integrates to \(y\), and the right side integrates to \(x^2 + C\) (where \(C\) is the constant of integration):
\[ y = x^2 + C \]

### Step 4: Use the initial condition
We know that the curve passes through the origin, which means when \(x = 0\), \(y = 0\). We can use this information to find the constant \(C\):
\[ 0 = 0^2 + C \]
This simplifies to:
\[ 0 = C \]

### Step 5: Write the final equation
Now that we have found \(C\), we can substitute it back into the equation:
\[ y = x^2 + 0 \]
Thus, the equation of the curve is:
\[ y = x^2 \]

### Final Answer
The equation of the curve is:
\[ y = x^2 \]

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