The solution of the differential equation `y sin (x//y) dx= (x sin (x//y)-y) dy` satisfying `y(pi//4)=1` is

Options

A. `"cos"(x)/(y)= log_(e)y+ (1)/(sqrt2)`
B. `"sin"(x)/(y) = log_(e)y + (1)/(sqrt2)`
C. `"sin "(x)/(y) = log_(e)x -(1)/(sqrt2)`
D. `"cos" (x)/(y) = -log_(e)x (1)/(sqrt2)`

Answer

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

### Step 1: Rewrite the Differential Equation
First, we can rewrite the given differential equation in a more manageable form:
\[ y \sin\left(\frac{x}{y}\right) dx - \left(x \sin\left(\frac{x}{y}\right) - y\right) dy = 0 \]

### Step 2: Change of Variables
To simplify the equation, we can use the substitution \( v = \frac{x}{y} \), which implies that \( x = vy \). Differentiating both sides with respect to \( y \):
\[ dx = v dy + y dv \]

### Step 3: Substitute into the Equation
Substituting \( x = vy \) and \( dx = v dy + y dv \) into the original equation:
\[ y \sin(v)(v dy + y dv) = (vy \sin(v) - y) dy \]
Expanding this gives:
\[ y \sin(v) v dy + y^2 \sin(v) dv = (vy \sin(v) - y) dy \]

### Step 4: Rearranging Terms
Rearranging the equation, we have:
\[ y^2 \sin(v) dv = -y dy \]
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ y \sin(v) dv = -dy \]

### Step 5: Integrate Both Sides
Now we can integrate both sides:
\[ \int y \sin(v) dv = -\int dy \]

### Step 6: Solve the Integrals
The left side can be integrated with respect to \( v \):
\[ y \left(-\cos(v)\right) = -y + C \]
Thus, we have:
\[ y \cos(v) = y + C \]

### Step 7: Substitute Back for \( v \)
Substituting back \( v = \frac{x}{y} \):
\[ y \cos\left(\frac{x}{y}\right) = y + C \]

### Step 8: Apply the Initial Condition
Using the initial condition \( y\left(\frac{\pi}{4}\right) = 1 \):
\[ 1 \cos\left(\frac{\frac{\pi}{4}}{1}\right) = 1 + C \]
Calculating \( \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \):
\[ \frac{1}{\sqrt{2}} = 1 + C \implies C = \frac{1}{\sqrt{2}} - 1 \]

### Step 9: Final Solution
Substituting \( C \) back into the equation gives the final solution:
\[ y \cos\left(\frac{x}{y}\right) = y + \left(\frac{1}{\sqrt{2}} - 1\right) \]