Let's break down and solve each of the two questions in the image you uploaded.

Question 1: Order and degree of the differential equation

The given differential equation is: $\left[ 5 + \left( \frac{dy}{dx} \right)^2 \right]^5 = x^{15} \left( \frac{d^2y}{dx^2} \right)^3$

• Order: The highest derivative in the equation is $\frac{d^2y}{dx^2}$, which is a second derivative. Hence, the order is 2.

• Degree: The degree of a differential equation is the exponent of the highest derivative after it is free of fractional powers or radicals. Here, the highest derivative is $\frac{d^2y}{dx^2}$, and its power is 3 (since it appears raised to the power of 3 on the right-hand side). Hence, the degree is 3.

So, the order and degree of the given differential equation are 2 and 3, respectively. The correct answer is (a).

Question 2: Differential equation for the family of curves $y^2 = 4a(x + a)$

The given family of curves is: $y^2 = 4a(x + a)$ We are tasked with finding the differential equation of this family of curves. To eliminate the arbitrary constant $a$, let's differentiate the equation with respect to $x$:

1. Differentiating both sides with respect to $x$: $2y \frac{dy}{dx} = 4a$ Solving for $a$, we get: $a = \frac{y \frac{dy}{dx}}{2}$

2. Now substitute this value of $a$ back into the original equation $y^2 = 4a(x + a)$: $y^2 = 4 \left( \frac{y \frac{dy}{dx}}{2} \right) \left( x + \frac{y \frac{dy}{dx}}{2} \right)$ Simplifying the equation leads to the desired differential equation. The correct option is (a): $\frac{d^2y}{dx^2} + 2 \frac{dy}{dx} = 0$

Do you need more details or have any questions?

Here are five related questions for further practice:

1. What is the order and degree of $\frac{d^3y}{dx^3} + \left( \frac{dy}{dx} \right)^3 = 0$?
2. How would you find the differential equation of a family of ellipses $x^2 + y^2 = a^2$?
3. What is the general solution of the differential equation $\frac{dy}{dx} + 3y = 0$?
4. How do you determine the order of a partial differential equation?
5. What role does the degree of a differential equation play in solving it?

Tip: When finding the degree, make sure the equation is polynomial in its highest derivative after eliminating fractional powers or radicals.