The solution of the differential equation `(xy^4 + y) dx-x dy = 0,` is

Options

A. `4x ^4 y^3 + 3x ^3 = cy ^3`
B. `3x ^3 y^4 +4x^3=cy^3`
C. `3x^4 y^3 + 4x ^3 = cy ^3`
D. none of these

Answer

To solve the differential equation \( (xy^4 + y) dx - x dy = 0 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given differential equation:
\[ (xy^4 + y) dx - x dy = 0 \]
Rearranging gives us:
\[ (xy^4 + y) dx = x dy \]

### Step 2: Dividing by \( dx \)
Next, we divide the entire equation by \( dx \):
\[ xy^4 + y = x \frac{dy}{dx} \]

### Step 3: Dividing by \( x \)
Now, we divide both sides by \( x \):
\[ \frac{dy}{dx} = \frac{y^4 + \frac{y}{x}}{x} \]

### Step 4: Dividing by \( y^4 \)
Next, we divide the equation by \( y^4 \):
\[ \frac{1}{y^4} \frac{dy}{dx} = \frac{1}{x} + \frac{1}{y^3} \]

### Step 5: Substituting \( v \)
Let \( v = -\frac{1}{y^3} \). Then, differentiating gives:
\[ \frac{dv}{dx} = 3 \frac{1}{y^4} \frac{dy}{dx} \]
Thus, we can express \( \frac{1}{y^4} \frac{dy}{dx} \) as:
\[ \frac{1}{y^4} \frac{dy}{dx} = \frac{1}{3} \frac{dv}{dx} \]

### Step 6: Substituting in the Equation
Substituting this back into our equation gives:
\[ \frac{1}{3} \frac{dv}{dx} = \frac{1}{x} + v \]
Multiplying through by 3 results in:
\[ \frac{dv}{dx} + \frac{3}{x} v = 3 \]

### Step 7: Finding the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int \frac{3}{x} dx} = e^{3 \ln |x|} = |x|^3 \]

### Step 8: Solving the Differential Equation
Multiplying the entire equation by the integrating factor:
\[ |x|^3 \frac{dv}{dx} + 3 |x|^2 v = 3 |x|^3 \]
This can be rewritten as:
\[ \frac{d}{dx} (|x|^3 v) = 3 |x|^3 \]
Integrating both sides:
\[ |x|^3 v = \frac{3}{4} |x|^4 + C \]

### Step 9: Substituting Back for \( v \)
Substituting back for \( v \):
\[ |x|^3 \left(-\frac{1}{y^3}\right) = \frac{3}{4} |x|^4 + C \]
This simplifies to:
\[ -\frac{|x|^3}{y^3} = \frac{3}{4} |x|^4 + C \]

### Step 10: Rearranging to Find \( y \)
Rearranging gives:
\[ y^3 = -\frac{4 |x|^3}{3|x|^4 + 4C} \]
Thus, we can express \( y \) as:
\[ y = \left(-\frac{4 |x|^3}{3|x|^4 + 4C}\right)^{1/3} \]

### Final Answer
The solution of the differential equation is:
\[ y^3 = \frac{4}{3} x^4 + C \]