The general solution of the DE ` (dy)/(dx) + (y)/(x) = x ^(2) ` is

Options

A. `xy = x ^(4) + C `
B. `4xy = x ^(4) + C `
C. ` 3xy = x ^(3) + C `
D. none of these

Explanation

To solve the differential equation \(\frac{dy}{dx} + \frac{y}{x} = x^2\), we will follow the standard method for linear differential equations.

### Step 1: Identify \(p(x)\) and \(q(x)\)
The given equation can be rewritten in the standard form:
\[ \frac{dy}{dx} + p(x)y = q(x) \]
Here, we have:
- \(p(x) = \frac{1}{x}\)
- \(q(x) = x^2\)

### Step 2: Find the Integrating Factor
The integrating factor \(I(x)\) is given by:
\[ I(x) = e^{\int p(x) \, dx} = e^{\int \frac{1}{x} \, dx} \]
Calculating the integral:
\[ \int \frac{1}{x} \, dx = \ln |x| \]
Thus, the integrating factor becomes:
\[ I(x) = e^{\ln |x|} = |x| \]
Since we are generally interested in positive \(x\) in this context, we can take:
\[ I(x) = x \]

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

### Step 4: Rewrite the Left Side as a Derivative
The left side can be rewritten as the derivative of a product:
\[ \frac{d}{dx}(xy) = x^3 \]

### Step 5: Integrate Both Sides
Now, integrate both sides with respect to \(x\):
\[ \int \frac{d}{dx}(xy) \, dx = \int x^3 \, dx \]
This gives:
\[ xy = \frac{x^4}{4} + C \]
where \(C\) is the constant of integration.

### Step 6: Solve for \(y\)
Finally, solve for \(y\):
\[ y = \frac{x^4}{4x} + \frac{C}{x} = \frac{x^3}{4} + \frac{C}{x} \]

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

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