Step 1: Understanding the Concept:
To ascertain the order and degree of a differential equation representing a family of curves, we must first formulate the general equation of that family. Since we are dealing with a family of tangent lines to a specified fixed parabola, the equation of any such line will depend on a single arbitrary parameter (like its slope). Eliminating this single parameter will yield a first-order differential equation. Analyzing this resulting equation gives us the order and degree.
Step 2: Key Formula or Approach:
1. Formulate the general equation of a tangent to the parabola $x^2 = 4ay$. A standard form is $y = mx - am^2$, where $m$ is the slope parameter.
2. Differentiate the equation with respect to $x$ to find a relation for the parameter $m$.
3. Substitute the expression for $m$ back into the tangent equation to eliminate it entirely.
4. Identify Order (highest derivative) and Degree (power of highest derivative when polynomial in derivatives).
Step 3: Detailed Explanation:
The given equation of the parabola is $x^2 = 4y$. By comparing it with the standard form $x^2 = 4ay$, we determine that $a = 1$.
The equation of any tangent line to this parabola can be expressed using its slope $m$ as:
$y = mx - am^2$
Substituting $a = 1$, we get the family of tangents:
$y = mx - m^2$ --- (Equation 1)
Here, $m$ is the single arbitrary parameter. To form the differential equation, we need to eliminate $m$.
Differentiate Equation 1 with respect to $x$:
$\frac{dy}{dx} = m \cdot 1 - 0$
So, we find that the parameter $m$ is exactly equal to the derivative: $m = \frac{dy}{dx}$.
Now, substitute this value back into Equation 1:
$y = \left(\frac{dy}{dx}\right)x - \left(\frac{dy}{dx}\right)^2$
Rearrange the equation into a standard polynomial form with respect to the derivatives:
$\left(\frac{dy}{dx}\right)^2 - x\left(\frac{dy}{dx}\right) + y = 0$
Let's analyze this final differential equation:
- Order: The highest order derivative present in the equation is $\frac{dy}{dx}$, which is a first derivative. Thus, the order is 1.
- Degree: The differential equation is a polynomial equation in its derivatives. The highest power to which the highest order derivative ($\frac{dy}{dx}$) is raised is 2. Thus, the degree is 2.
Step 4: Final Answer:
The order is 1 and the degree is 2.