The solution of the differential equation `x(dy)/(dx)+y=x^(3)y^(6)`, is

Options

A. `x^(7)=5y^(5)+Cx^(2)y^(5)`
B. `2x^(7)=5y^(5)+Cx^(2)y^(5)`
C. `5x^(7)=2y^(5)+Cx^(2)y^(5)`
D. `2x^(7)=5y^(5)+Cx^(5)y^(2)`

Explanation

To solve the differential equation \( x \frac{dy}{dx} + y = x^3 y^6 \), we will follow a systematic approach.

### Step 1: Rewrite the Equation
We start with the given equation:
\[ x \frac{dy}{dx} + y = x^3 y^6 \]
We can rearrange this to isolate the derivative:
\[ x \frac{dy}{dx} = x^3 y^6 - y \]

### Step 2: Divide by \( y^6 \)
To simplify, we divide the entire equation by \( y^6 \):
\[ \frac{dy}{dx} + \frac{y}{x} = x^2 \]
This is now in the standard form of a linear differential equation.

### Step 3: Identify \( p(x) \) and \( q(x) \)
From the equation \( \frac{dy}{dx} + \frac{1}{x} y = x^2 \):
- \( p(x) = \frac{1}{x} \)
- \( q(x) = x^2 \)

### Step 4: Find the Integrating Factor
The integrating factor \( \mu(x) \) is given by:
\[ \mu(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} = e^{\ln |x|} = |x| \]
Since \( x \) is positive in the context of this problem, we can simply use \( \mu(x) = x \).

### Step 5: Multiply the Equation by the Integrating Factor
Now we multiply the entire differential equation by \( x \):
\[ x \frac{dy}{dx} + y = x^3 \]

### Step 6: Rewrite the Left Side
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(xy) = x^3 \]

### Step 7: Integrate Both Sides
Now we integrate both sides with respect to \( x \):
\[ \int \frac{d}{dx}(xy) \, dx = \int x^3 \, dx \]
This gives:
\[ xy = \frac{x^4}{4} + C \]

### Step 8: Solve for \( y \)
Now we can solve for \( y \):
\[ y = \frac{x^4}{4x} + \frac{C}{x} = \frac{x^3}{4} + \frac{C}{x} \]

### Final Solution
Thus, the solution to the differential equation is:
\[ y = \frac{x^3}{4} + \frac{C}{x} \]