The equation of one of the curves whose slope at any point is equal to `y+2x and y(0)=0` is

Options

A. `y2(e^(x)+x-1)`
B. `y=2 (e^(x)-x-1)`
C. `y=2(e^(x)-x+1)`
D. `y=2(e^(x)+x+1)`

Answer

To solve the given differential equation where the slope at any point is equal to \( y + 2x \) and the initial condition is \( y(0) = 0 \), we will follow these steps:

### Step 1: Write the differential equation
The slope of the curve is given by:
\[ \frac{dy}{dx} = y + 2x \]

### Step 2: Rearrange the equation
Rearranging gives:
\[ \frac{dy}{dx} - y = 2x \]
This is a first-order linear differential equation in the standard form:
\[ \frac{dy}{dx} + P(y) = Q(x) \]
where \( P(y) = -1 \) and \( Q(x) = 2x \).

### Step 3: Find the integrating factor
The integrating factor \( IF \) is given by:
\[ IF = e^{\int P(x) dx} = e^{\int -1 \, dx} = e^{-x} \]

### Step 4: Multiply the entire equation by the integrating factor
Multiplying the entire differential equation by the integrating factor:
\[ e^{-x} \frac{dy}{dx} - e^{-x} y = 2x e^{-x} \]

### Step 5: Rewrite the left side as a derivative
The left side can be rewritten as:
\[ \frac{d}{dx}(y e^{-x}) = 2x e^{-x} \]

### Step 6: Integrate both sides
Integrating both sides with respect to \( x \):
\[ \int \frac{d}{dx}(y e^{-x}) \, dx = \int 2x e^{-x} \, dx \]
The left side simplifies to:
\[ y e^{-x} = \int 2x e^{-x} \, dx \]
To solve the right side, we can use integration by parts. Let:
- \( u = 2x \) and \( dv = e^{-x} dx \)
Then:
- \( du = 2 dx \) and \( v = -e^{-x} \)

Using integration by parts:
\[ \int 2x e^{-x} \, dx = -2x e^{-x} - \int -2 e^{-x} \, dx = -2x e^{-x} + 2 e^{-x} + C \]
Thus:
\[ y e^{-x} = -2x e^{-x} + 2 e^{-x} + C \]

### Step 7: Solve for \( y \)
Multiplying through by \( e^{x} \):
\[ y = -2x + 2 + Ce^{x} \]

### Step 8: Apply the initial condition
Using the initial condition \( y(0) = 0 \):
\[ 0 = -2(0) + 2 + Ce^{0} \]
This simplifies to:
\[ 0 = 2 + C \implies C = -2 \]

### Step 9: Write the final solution
Substituting \( C \) back into the equation:
\[ y = -2x + 2 - 2e^{x} \]

Thus, the equation of the curve is:
\[ y = -2x + 2 - 2e^{x} \]