Solve the following differential equations
`(dy)/(dx)=(y)/(x)+tan (y/x)`.

Answer

To solve the differential equation

\[ \frac{dy}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right), \]

we will use the substitution \( v = \frac{y}{x} \). This implies that \( y = vx \).

### Step 1: Differentiate \( y \)
Using the product rule, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = v + x \frac{dv}{dx}. \]

### Step 2: Substitute into the original equation
Now, we substitute \( \frac{dy}{dx} \) and \( v \) into the original differential equation:

\[ v + x \frac{dv}{dx} = \frac{y}{x} + \tan\left(\frac{y}{x}\right). \]

Since \( \frac{y}{x} = v \), we have:

\[ v + x \frac{dv}{dx} = v + \tan(v). \]

### 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:

\[ \int \cot(v) \, dv = \int \frac{dx}{x}. \]

The integrals yield:

\[ \log|\sin(v)| = \log|x| + C, \]

where \( C \) is the constant of integration.

### Step 6: Exponentiate both sides
Exponentiating both sides gives:

\[ |\sin(v)| = e^C |x| = C' |x|, \]

where \( C' = e^C \).

### Step 7: Substitute back for \( v \)
Recall that \( v = \frac{y}{x} \), thus:

\[ |\sin\left(\frac{y}{x}\right)| = C' |x|. \]

### Step 8: Final solution
We can rewrite this as:

\[ \sin\left(\frac{y}{x}\right) = Cx, \]

where \( C \) is a constant.

Thus, the solution to the differential equation is:

\[ \sin\left(\frac{y}{x}\right) = Cx. \]

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