`(dy)/(dx)=(y)/(x)+tan((y)/(x))`

Options

A. `cx=sin((x)/(y))`
B. `cx-sin((y)/(x))`
C. `cx=cos((y)/(x))`
D. `(y)/(x).cos((y)/(x))=c`

Answer

To solve the differential equation \(\frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right)\), we can follow these steps:

### Step 1: Substitution
Let \(y = vx\), where \(v\) is a function of \(x\). Then, we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = v + x\frac{dv}{dx} \]

### Step 2: Substitute into the Equation
Now, substitute \(y = vx\) into the original differential equation:
\[ v + x\frac{dv}{dx} = \frac{vx}{x} + \tan\left(\frac{vx}{x}\right) \]
This simplifies to:
\[ v + x\frac{dv}{dx} = v + \tan(v) \]

### Step 3: Simplify the Equation
Subtract \(v\) from both sides:
\[ x\frac{dv}{dx} = \tan(v) \]

### Step 4: Separate Variables
Now, we can separate the variables:
\[ \frac{dv}{\tan(v)} = \frac{dx}{x} \]

### Step 5: Integrate Both Sides
Integrate both sides:
\[ \int \frac{dv}{\tan(v)} = \int \frac{dx}{x} \]
The left side can be rewritten as:
\[ \int \cot(v) \, dv = \int \frac{dx}{x} \]

### Step 6: Integrate
The integrals yield:
\[ \log|\sin(v)| = \log|x| + C \]
where \(C\) is the constant of integration.

### Step 7: Exponentiate Both Sides
Exponentiating both sides gives:
\[ |\sin(v)| = k|x| \quad \text{where } k = e^C \]

### Step 8: Substitute Back for \(v\)
Recall that \(v = \frac{y}{x}\), so we substitute back:
\[ |\sin\left(\frac{y}{x}\right)| = k|x| \]

### Final Solution
Thus, the solution to the differential equation is:
\[ \sin\left(\frac{y}{x}\right) = Cx \]
where \(C\) is a constant.

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