Solve the following differential equations :
`x^(2)dy=(2xy+y^(2))dx`, given that `y=1` when `x=1`.

Answer

To solve the differential equation \( x^2 dy = (2xy + y^2) dx \) with the initial condition \( y = 1 \) when \( x = 1 \), we can follow these steps:

### Step 1: Rewrite the Differential Equation
We start with the given equation:
\[ x^2 dy = (2xy + y^2) dx \]
Now, we can separate the variables by dividing both sides by \( x^2 \) and \( (2xy + y^2) \):
\[ \frac{dy}{dx} = \frac{2xy + y^2}{x^2} \]

### Step 2: Identify the Homogeneous Form
The equation can be rewritten as:
\[ \frac{dy}{dx} = \frac{y(2x + y)}{x^2} \]
This is a homogeneous differential equation because it can be expressed in the form \( \frac{dy}{dx} = f\left(\frac{y}{x}\right) \).

### Step 3: Substitute \( y = vx \)
Let \( y = vx \), where \( v \) is a function of \( x \). Then, \( dy = v dx + x dv \). Substituting these into the equation gives:
\[ v dx + x dv = \frac{(2x(vx) + (vx)^2)}{x^2} dx \]
Simplifying the right side:
\[ v dx + x dv = \frac{2v + v^2}{x} dx \]

### Step 4: Rearranging the Equation
Now, we can rearrange the equation:
\[ x dv = \frac{2v + v^2 - v}{x} dx \]
This simplifies to:
\[ x dv = \frac{(v + 2)v}{x} dx \]
Dividing both sides by \( v + 2 \):
\[ \frac{dv}{v + 2} = \frac{dx}{x} \]

### Step 5: Integrate Both Sides
Now we integrate both sides:
\[ \int \frac{dv}{v + 2} = \int \frac{dx}{x} \]
The integrals yield:
\[ \ln |v + 2| = \ln |x| + C \]

### Step 6: Solve for \( v \)
Exponentiating both sides gives:
\[ |v + 2| = k|x| \quad \text{where } k = e^C \]
Thus,
\[ v + 2 = kx \quad \text{or} \quad v = kx - 2 \]

### Step 7: Substitute Back for \( y \)
Recall that \( v = \frac{y}{x} \), so:
\[ \frac{y}{x} = kx - 2 \]
Multiplying through by \( x \):
\[ y = kx^2 - 2x \]

### Step 8: Apply the Initial Condition
Using the initial condition \( y = 1 \) when \( x = 1 \):
\[ 1 = k(1)^2 - 2(1) \]
This simplifies to:
\[ 1 = k - 2 \implies k = 3 \]

### Step 9: Write the Final Solution
Substituting \( k \) back into the equation for \( y \):
\[ y = 3x^2 - 2x \]

### Final Answer
The solution to the differential equation is:
\[ y = 3x^2 - 2x \]