If the curve satisfying differential equation `xdy=(y+x^(3))dx` passes through `(1, 1),` then the equation to the curve is

Options

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

Answer

To solve the given differential equation \( x \, dy = (y + x^3) \, dx \) and find the equation of the curve that passes through the point \( (1, 1) \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ x \, dy = (y + x^3) \, dx \]
Dividing both sides by \( x \) gives:
\[ dy = \left( \frac{y}{x} + x^2 \right) dx \]

### Step 2: Writing in Standard Form
We can rearrange this to the standard form of a linear differential equation:
\[ \frac{dy}{dx} - \frac{y}{x} = x^2 \]

### Step 3: Identifying \( p(x) \) and \( q(x) \)
From the standard form \( \frac{dy}{dx} + p(x) y = q(x) \), we identify:
- \( p(x) = -\frac{1}{x} \)
- \( q(x) = x^2 \)

### Step 4: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int -\frac{1}{x} \, dx} = e^{-\ln |x|} = \frac{1}{x} \]

### Step 5: Multiplying the Equation by the Integrating Factor
Multiplying the entire differential equation by the integrating factor:
\[ \frac{1}{x} \frac{dy}{dx} - \frac{y}{x^2} = x \]
This simplifies to:
\[ \frac{d}{dx} \left( \frac{y}{x} \right) = x \]

### Step 6: Integrating Both Sides
Integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx} \left( \frac{y}{x} \right) \, dx = \int x \, dx \]
This gives:
\[ \frac{y}{x} = \frac{x^2}{2} + C \]

### Step 7: Solving for \( y \)
Multiplying through by \( x \):
\[ y = \frac{x^3}{2} + Cx \]

### Step 8: Using the Initial Condition
We know the curve passes through the point \( (1, 1) \):
\[ 1 = \frac{1^3}{2} + C(1) \]
This simplifies to:
\[ 1 = \frac{1}{2} + C \implies C = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 9: Final Equation of the Curve
Substituting \( C \) back into the equation for \( y \):
\[ y = \frac{x^3}{2} + \frac{1}{2} x \]
Multiplying through by 2 to eliminate the fraction:
\[ 2y = x^3 + x \]

### Final Answer
The equation of the curve is:
\[ 2y = x^3 + x \]

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