Find the differential equation of an ellipse with major and minor axes 2a and 2b respectively.

Answer

To find the differential equation of an ellipse with major and minor axes of lengths \(2a\) and \(2b\) respectively, we start with the standard equation of the ellipse:

### Step 1: Write the equation of the ellipse
The equation of the ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]

### Step 2: Differentiate the equation with respect to \(x\)
We differentiate the equation implicitly with respect to \(x\):
\[ \frac{d}{dx}\left(\frac{x^2}{a^2}\right) + \frac{d}{dx}\left(\frac{y^2}{b^2}\right) = \frac{d}{dx}(1) \]
This gives us:
\[ \frac{2x}{a^2} + \frac{2y}{b^2} \frac{dy}{dx} = 0 \]

### Step 3: Rearrange the equation
Rearranging the equation, we can express it as:
\[ \frac{2y}{b^2} \frac{dy}{dx} = -\frac{2x}{a^2} \]
Dividing through by 2:
\[ \frac{y}{b^2} \frac{dy}{dx} = -\frac{x}{a^2} \]

### Step 4: Differentiate again to eliminate constants
Now we differentiate this equation again with respect to \(x\):
\[ \frac{d}{dx}\left(\frac{y}{b^2} \frac{dy}{dx}\right) = \frac{d}{dx}\left(-\frac{x}{a^2}\right) \]
Using the product rule on the left side:
\[ \frac{1}{b^2} \left(\frac{dy}{dx} \cdot \frac{dy}{dx} + y \cdot \frac{d^2y}{dx^2}\right) = -\frac{1}{a^2} \]

### Step 5: Multiply through by \(b^2\)
Multiplying through by \(b^2\) gives:
\[ \frac{dy}{dx}^2 + y \frac{d^2y}{dx^2} = -\frac{b^2}{a^2} \]

### Step 6: Rearranging the equation
Rearranging the equation, we have:
\[ y \frac{d^2y}{dx^2} + \frac{dy}{dx}^2 + \frac{b^2}{a^2} = 0 \]

### Step 7: Final form of the differential equation
This can be expressed as:
\[ y \frac{d^2y}{dx^2} + \frac{dy}{dx}^2 = -\frac{b^2}{a^2} \]

This is the required differential equation of the ellipse.

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