The solution of the differential equation `y(dy)/(dx)=x-1` satisfying y(1) = 1, is

Options

A. `y^(2)=x^(2)-2x+2`
B. `y^(2)=2x^(2)-x-1`
C. `y=x^(2)-2x+2`
D. none of these

Answer

To solve the differential equation \( y \frac{dy}{dx} = x - 1 \) with the initial condition \( y(1) = 1 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ y \frac{dy}{dx} = x - 1 \]
We can rearrange this to separate the variables:
\[ y \, dy = (x - 1) \, dx \]

### Step 2: Integrating Both Sides
Next, we integrate both sides:
\[ \int y \, dy = \int (x - 1) \, dx \]
The left side integrates to:
\[ \frac{y^2}{2} \]
The right side integrates to:
\[ \frac{x^2}{2} - x + C \]
Thus, we have:
\[ \frac{y^2}{2} = \frac{x^2}{2} - x + C \]

### Step 3: Simplifying the Equation
To eliminate the fraction, we multiply through by 2:
\[ y^2 = x^2 - 2x + 2C \]

### Step 4: Applying the Initial Condition
Now we use the initial condition \( y(1) = 1 \) to find the constant \( C \):
Substituting \( x = 1 \) and \( y = 1 \):
\[ 1^2 = 1^2 - 2(1) + 2C \]
This simplifies to:
\[ 1 = 1 - 2 + 2C \]
\[ 1 = -1 + 2C \]
Adding 1 to both sides gives:
\[ 2 = 2C \]
Dividing by 2, we find:
\[ C = 1 \]

### Step 5: Final Equation
Substituting \( C = 1 \) back into the equation:
\[ y^2 = x^2 - 2x + 2(1) \]
This simplifies to:
\[ y^2 = x^2 - 2x + 2 \]

### Conclusion
Thus, the solution to the differential equation \( y \frac{dy}{dx} = x - 1 \) satisfying \( y(1) = 1 \) is:
\[ y^2 = x^2 - 2x + 2 \]