The differential equation `x(dy)/(dx)+(3)/((dy)/(dx))=y^(2)`

Options

A. is of order 1
B. is of degree 2
C. is linear
D. is non-linear

Answer

To solve the differential equation \( x \frac{dy}{dx} + \frac{3}{\frac{dy}{dx}} = y^2 \), we will follow these steps:

### Step 1: Multiply through by \(\frac{dy}{dx}\)
To eliminate the fraction, we multiply the entire equation by \(\frac{dy}{dx}\):

\[ x \left( \frac{dy}{dx} \right)^2 + 3 = y^2 \frac{dy}{dx} \]

### Step 2: Rearrange the equation
Now, we can rearrange the equation to isolate the terms involving \(\frac{dy}{dx}\):

\[ x \left( \frac{dy}{dx} \right)^2 - y^2 \frac{dy}{dx} + 3 = 0 \]

### Step 3: Treat this as a quadratic equation
This is a quadratic equation in terms of \(\frac{dy}{dx}\). We can identify \(A\), \(B\), and \(C\) as follows:

- \(A = x\)
- \(B = -y^2\)
- \(C = 3\)

### Step 4: Apply the quadratic formula
Using the quadratic formula \(\frac{dy}{dx} = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\):

\[ \frac{dy}{dx} = \frac{-(-y^2) \pm \sqrt{(-y^2)^2 - 4 \cdot x \cdot 3}}{2x} \]

This simplifies to:

\[ \frac{dy}{dx} = \frac{y^2 \pm \sqrt{y^4 - 12x}}{2x} \]

### Step 5: Separate variables
Now we have two potential expressions for \(\frac{dy}{dx}\). We can separate variables for each case. Let's consider the case with the positive sign:

\[ \frac{dy}{dx} = \frac{y^2 + \sqrt{y^4 - 12x}}{2x} \]

### Step 6: Integrate
To find \(y\) as a function of \(x\), we would need to integrate both sides. This step may require specific techniques depending on the form of the equation.

### Step 7: General solution
After integrating, we will arrive at the general solution of the differential equation.

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