Find the particular solution of the differential equation `{xsin^(2)(y/x)-y}dx+xdy=0`, it being given that `y=pi/4` when `x=1`.

Answer

To solve the differential equation \( x \sin^2\left(\frac{y}{x}\right) - y \, dx + x \, dy = 0 \) with the initial condition \( y = \frac{\pi}{4} \) when \( x = 1 \), we will follow these steps:

### Step 1: Rewrite the Differential Equation
We can rearrange the given equation into the form:
\[ \frac{dy}{dx} = \frac{y - x \sin^2\left(\frac{y}{x}\right)}{x} \]

### Step 2: Check for Homogeneity
The equation is homogeneous because if we replace \( x \) with \( \lambda x \) and \( y \) with \( \lambda y \), the equation remains unchanged. Thus, we can use the substitution \( y = vx \), where \( v \) is a function of \( x \).

### Step 3: Substitute \( y = vx \)
Substituting \( y = vx \) gives us:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Substituting this into the differential equation, we have:
\[ v + x \frac{dv}{dx} = \frac{vx - x \sin^2(v)}{x} \]
This simplifies to:
\[ v + x \frac{dv}{dx} = v - \sin^2(v) \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ x \frac{dv}{dx} = -\sin^2(v) \]
This can be rewritten as:
\[ \frac{dv}{\sin^2(v)} = -\frac{dx}{x} \]

### Step 5: Integrate Both Sides
Integrating both sides:
\[ \int \frac{dv}{\sin^2(v)} = -\int \frac{dx}{x} \]
The left side integrates to:
\[ -\cot(v) = -\log|x| + C \]

### Step 6: Solve for \( C \)
Rearranging gives:
\[ \cot(v) = \log|x| + C \]
Substituting back \( v = \frac{y}{x} \):
\[ \cot\left(\frac{y}{x}\right) = \log|x| + C \]

### Step 7: Apply Initial Condition
Using the initial condition \( y = \frac{\pi}{4} \) when \( x = 1 \):
\[ \cot\left(\frac{\pi}{4}\right) = \log(1) + C \]
Since \( \cot\left(\frac{\pi}{4}\right) = 1 \) and \( \log(1) = 0 \):
\[ 1 = 0 + C \implies C = 1 \]

### Step 8: Final Solution
Substituting \( C \) back into the equation:
\[ \cot\left(\frac{y}{x}\right) = \log|x| + 1 \]

This is the particular solution of the differential equation.