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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.

1. The order and degree of the differential equation [5 + (dy/dx)^2]^5 = x^15 (d^2y/dx^2)^3. 2. The differential equation of the family of curves y^2 = 4a(x + a), where a is an arbitrary constant.

Explore how to solve differential equations, determine their order and degree, and find the differential equation of a family of curves. Learn the methods used to handle complex calculus problems, including eliminating arbitrary constants and using differentiation.

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