If `(dy)/(dx)-(y-3x)/(ln(y-3x))=3` then

Options

A. `(ln(y-3x))/2=x+c`
B. `(ln^2(y-3x))/2=x+c`
C. `(ln(y-3x))/2=x^2+c`
D. `(ln^2(y-3x))/2=x^2+c`

Answer

To solve the differential equation

\[ \frac{dy}{dx} - \frac{y - 3x}{\ln(y - 3x)} = 3, \]

we will follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to isolate \(\frac{dy}{dx}\):

\[ \frac{dy}{dx} = \frac{y - 3x}{\ln(y - 3x)} + 3. \]

**Hint:** Isolate \(\frac{dy}{dx}\) to simplify the equation.

### Step 2: Expressing the Left Side
Notice that we can express the left side as the derivative of a function. We can rewrite the equation as:

\[ \frac{dy}{dx} - 3 = \frac{y - 3x}{\ln(y - 3x)}. \]

This suggests that we can express the left side as the derivative of \(y - 3x\):

\[ \frac{d}{dx}(y - 3x) = \frac{y - 3x}{\ln(y - 3x)}. \]

**Hint:** Recognize that the left side can be expressed as a derivative.

### Step 3: Substituting Variables
Let \(t = \ln(y - 3x)\). Then, differentiating \(t\) gives us:

\[ dt = \frac{1}{y - 3x} \left(\frac{dy}{dx} - 3\right)dx. \]

Substituting this into the equation gives:

\[ dt = \frac{y - 3x}{t}dx. \]

**Hint:** Use substitution to simplify the equation.

### Step 4: Integrating Both Sides
Now we can integrate both sides. The left side becomes:

\[ \int dt = t, \]

and the right side becomes:

\[ \int \frac{y - 3x}{t}dx. \]

This leads to:

\[ t = \frac{(y - 3x)^2}{2} + C, \]

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

**Hint:** Remember to integrate both sides carefully.

### Step 5: Back Substituting
Now substitute back \(t = \ln(y - 3x)\):

\[ \ln(y - 3x) = \frac{(y - 3x)^2}{2} + C. \]

**Hint:** Substitute back to express the solution in terms of the original variables.

### Final Step: Rearranging the Solution
The final form of the solution can be expressed as:

\[ \ln^2(y - 3x) = 2(x + C). \]

**Hint:** Rearranging the equation can help in identifying the correct form of the solution.

### Conclusion
Thus, the solution to the differential equation is:

\[ \ln^2(y - 3x) = 2(x + C). \]

This matches with option number 2 in the given choices.