The solution of differential equation `4xdy - ydx = x^(2) dy` is

Options

A. `y= Ce^(4x) `
B. `(x-4)y^(4) = Cx `
C. `x^(4) - y^(4) = Cxy`
D. None of these

Answer

To solve the differential equation \( 4x \, dy - y \, dx = x^2 \, dy \), we can follow these steps:

### Step 1: Rearranging the Equation
Start by rearranging the equation to group the \( dy \) terms on one side and the \( dx \) terms on the other side.

\[ 4x \, dy - x^2 \, dy = y \, dx \]

This simplifies to:

\[ (4x - x^2) \, dy = y \, dx \]

### Step 2: Separating Variables
Next, we can separate the variables \( y \) and \( x \):

\[ \frac{dy}{y} = \frac{dx}{4x - x^2} \]

### Step 3: Integrating Both Sides
Now, we integrate both sides. The left side integrates to:

\[ \int \frac{dy}{y} = \ln |y| + C_1 \]

For the right side, we need to simplify the expression \( 4x - x^2 \):

\[ 4x - x^2 = - (x^2 - 4x) = - (x(x - 4)) \]

Thus, we can rewrite the integral as:

\[ \int \frac{dx}{4x - x^2} = \int \frac{dx}{- (x(x - 4))} \]

Using partial fractions, we can express this as:

\[ \frac{1}{- (x(x - 4))} = \frac{A}{x} + \frac{B}{x - 4} \]

Multiplying through by the denominator gives:

\[ 1 = -A(x - 4) - Bx \]

Setting up the equations for \( A \) and \( B \) leads to:

\[ A + B = 0 \quad \text{and} \quad -4A = 1 \]

Solving these gives \( A = -\frac{1}{4} \) and \( B = \frac{1}{4} \). Thus:

\[ \int \frac{dx}{4x - x^2} = -\frac{1}{4} \ln |x| + \frac{1}{4} \ln |x - 4| + C_2 \]

### Step 4: Combining Results
Combining the results from both integrals, we have:

\[ \ln |y| = -\frac{1}{4} \ln |x| + \frac{1}{4} \ln |x - 4| + C \]

### Step 5: Exponentiating Both Sides
Exponentiating both sides to eliminate the logarithm gives:

\[ |y| = e^C \cdot \left( \frac{|x - 4|^{1/4}}{|x|^{1/4}} \right) \]

Let \( K = e^C \), then:

\[ y = K \cdot \left( \frac{|x - 4|^{1/4}}{|x|^{1/4}} \right) \]

### Step 6: Final Form
To express the solution in a more standard form, we can write:

\[ y^4 = K \cdot \frac{x - 4}{x} \]

Thus, the final solution to the differential equation is:

\[ y^4 (4 - x) = Kx \]