If `xsin((y)/(x))dy = (y sin((y)/(x))-x)dx`and y(1) `= (pi)/(2)` then the value of `cos((y)/(e^(7)))` is __________.

Answer

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

### Step 1: Rewrite the Differential Equation
We start with the equation:
\[ x \sin\left(\frac{y}{x}\right) dy = (y \sin\left(\frac{y}{x}\right) - x) dx \]
We can rearrange this to express \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y \sin\left(\frac{y}{x}\right) - x}{x \sin\left(\frac{y}{x}\right)} \]

### Step 2: Substitute \( y = vx \)
Let \( v = \frac{y}{x} \), which implies \( y = vx \). Differentiating both sides with respect to \( x \) gives:
\[ \frac{dy}{dx} = v + x \frac{dv}{dx} \]
Substituting \( y = vx \) into the equation gives:
\[ \frac{dy}{dx} = \frac{vx \sin(v) - x}{x \sin(v)} = \frac{v \sin(v) - 1}{\sin(v)} \]
Thus, we have:
\[ v + x \frac{dv}{dx} = \frac{v \sin(v) - 1}{\sin(v)} \]

### Step 3: Rearranging the Equation
Rearranging gives:
\[ x \frac{dv}{dx} = \frac{v \sin(v) - 1}{\sin(v)} - v \]
This simplifies to:
\[ x \frac{dv}{dx} = \frac{v \sin(v) - 1 - v \sin(v)}{\sin(v)} = \frac{-1}{\sin(v)} \]

### Step 4: Separate Variables
We can separate the variables:
\[ \sin(v) dv = -\frac{dx}{x} \]

### Step 5: Integrate Both Sides
Integrating both sides gives:
\[ -\cos(v) = -\ln|x| + C \]
or
\[ \cos(v) = \ln|x| + C \]

### Step 6: Apply Initial Condition
Using the initial condition \( y(1) = \frac{\pi}{2} \), we have:
\[ v = \frac{y}{x} = \frac{\frac{\pi}{2}}{1} = \frac{\pi}{2} \]
Substituting \( x = 1 \) and \( v = \frac{\pi}{2} \):
\[ \cos\left(\frac{\pi}{2}\right) = \ln(1) + C \implies 0 = 0 + C \implies C = 0 \]
Thus, we have:
\[ \cos(v) = \ln|x| \]

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

### Step 8: Find \( \cos\left(\frac{y}{e^7}\right) \)
Now, we need to find \( \cos\left(\frac{y}{e^7}\right) \):
\[ \cos\left(\frac{y}{e^7}\right) = \ln|e^7| = 7 \]

### Final Answer
Thus, the value of \( \cos\left(\frac{y}{e^7}\right) \) is:
\[ \boxed{7} \]