Solve the following initial value problems and find the corresponding solution curves :
`x (x dy - y dx)= y dx, y(1)=1`.

Answer

To solve the initial value problem given by the differential equation \( x (x dy - y dx) = y dx \) with the initial condition \( y(1) = 1 \), we will follow these steps:

### Step 1: Rearranging the Equation
Starting with the equation:
\[ x (x dy - y dx) = y dx \]
we can expand and rearrange it:
\[ x^2 dy - xy dx = y dx \]
This simplifies to:
\[ x^2 dy = y dx + xy dx \]
or:
\[ x^2 dy = (y + xy) dx \]

### Step 2: Dividing by \( x^2 \)
Next, we divide both sides by \( x^2 \):
\[ dy = \frac{y + xy}{x^2} dx \]
This can be rewritten as:
\[ \frac{dy}{dx} = \frac{y + xy}{x^2} \]

### Step 3: Substituting \( y = vx \)
We will use the substitution \( y = vx \), where \( v \) is a function of \( x \). Then, we have:
\[ dy = v dx + x dv \]
Substituting this into our equation gives:
\[ v + x \frac{dv}{dx} = \frac{vx + x^2 v}{x^2} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = \frac{v}{x} + 1 \]

### Step 4: Rearranging the Equation
Rearranging the equation leads to:
\[ x \frac{dv}{dx} = \frac{v}{x} + 1 - v \]
or:
\[ x \frac{dv}{dx} = \frac{1 - v}{x} + 1 \]

### Step 5: Separating Variables
We can separate the variables:
\[ \frac{dv}{1 - v} = \frac{dx}{x^2} \]

### Step 6: Integrating Both Sides
Integrating both sides gives:
\[ \int \frac{dv}{1 - v} = \int \frac{dx}{x^2} \]
The left side integrates to:
\[ -\ln |1 - v| = -\frac{1}{x} + C \]

### Step 7: Solving for \( v \)
Exponentiating both sides results in:
\[ 1 - v = \frac{C}{e^{-\frac{1}{x}}} \]
Thus:
\[ v = 1 - \frac{C}{e^{-\frac{1}{x}}} \]

### Step 8: Substituting Back for \( y \)
Recalling that \( y = vx \), we have:
\[ y = x \left(1 - \frac{C}{e^{-\frac{1}{x}}}\right) \]

### Step 9: Applying the Initial Condition
Using the initial condition \( y(1) = 1 \):
\[ 1 = 1 \left(1 - \frac{C}{e^{-1}}\right) \]
This leads to:
\[ 1 - C e = 1 \implies C = 0 \]

### Final Solution
Thus, substituting \( C = 0 \) back into our equation gives:
\[ y = x \]

### Summary of Steps
1. Rearrange the differential equation.
2. Divide by \( x^2 \).
3. Substitute \( y = vx \).
4. Rearrange the equation.
5. Separate variables.
6. Integrate both sides.
7. Solve for \( v \).
8. Substitute back for \( y \).
9. Apply the initial condition to find \( C \).