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 can 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 terms involving \(\frac{dy}{dx}\):

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

### Step 3: Identify the form of the equation
This is a quadratic equation in terms of \(\frac{dy}{dx}\). We can express it in the standard quadratic form:

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

### Step 4: Apply the quadratic formula
To solve for \(\frac{dy}{dx}\), we can use the quadratic formula:

\[ \frac{dy}{dx} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \(a\), \(b\), and \(c\):

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

### Step 5: Simplify the expression
Now, simplify the expression under the square root:

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

### Step 6: Final form
Thus, the solution for the differential equation is:

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