Statement I The solution of `(ydx-xdy)cos((x)/(y))=ny^(2)dx " is " sin((x)/(y))=Ce^(nx)` Statement II Such type of differential equation can only be solved by the substitution x=vy.

Options

A. Statement I is true ,and Statement II is the correct explanation for Statement I.
B. Statement I is true, Statement II is true and Statement II is the correct explanation for Statment I
C. Statement I is true, Statement II is false.
D. Statement I is false, Statement II is true.

Answer

To solve the differential equation given by the statement `(ydx - xdy)cos(x/y) = ny^2dx`, we will follow a systematic approach.

### Step-by-Step Solution:

1. **Rearranging the Equation:**
Start with the given equation:
\[ (ydx - xdy) \cos\left(\frac{x}{y}\right) = ny^2dx \]
We can rearrange it by dividing both sides by \(y^2\):
\[ \frac{ydx - xdy}{y^2} \cos\left(\frac{x}{y}\right) = n dx \]

2. **Simplifying the Left Side:**
Notice that the left side can be rewritten as:
\[ \frac{1}{y^2}(ydx - xdy) = \frac{1}{y^2} \left( y \frac{dx}{dy} - x \right) dy \]
This suggests a substitution where \(v = \frac{x}{y}\), leading to \(x = vy\) and \(dx = v dy + y dv\).

3. **Substituting into the Equation:**
Substitute \(x = vy\) into the equation:
\[ (y(v dy + y dv) - vy dy) \cos(v) = ny^2 dy \]
Simplifying gives:
\[ (y^2 dv) \cos(v) = ny^2 dy \]
Dividing both sides by \(y^2\) (assuming \(y \neq 0\)):
\[ dv \cos(v) = n dy \]

4. **Integrating Both Sides:**
Now, integrate both sides:
\[ \int \cos(v) dv = \int n dy \]
The left side integrates to \(\sin(v)\) and the right side integrates to \(ny + C\):
\[ \sin(v) = ny + C \]

5. **Substituting Back:**
Recall that \(v = \frac{x}{y}\):
\[ \sin\left(\frac{x}{y}\right) = ny + C \]
Rearranging gives:
\[ \sin\left(\frac{x}{y}\right) = C e^{nx} \]
This confirms the solution provided in Statement I.

### Conclusion:
- **Statement I** is true as we have derived the solution correctly.
- **Statement II** is also true since this type of differential equation can indeed be solved using the substitution \(x = vy\).