Which of the following differential equation has the same order and degree?

Options

A. `(d^(4)y)/(dx^(4))+8((dy)/(dx))^(6)+5y=e^(x)`
B. `5((d^(3)y)/(dx^(3)))^(4)+8(1+(dy)/(dx))^(2)+5y=x^(8)`
C. `(1+((dy)/(dx))^(3))^(2/3)=4(d^(3)y)/(dx^(3))`
D. `y=x^(2)(dy)/(dx)+sqrt(1+((dy)/(dx))^(2)`

Answer

To determine which differential equation has the same order and degree, we will analyze each given equation step by step.

### Step-by-step Solution:

1. **Understanding Order and Degree**:
- The **order** of a differential equation is the highest derivative present in the equation.
- The **degree** is the power of the highest derivative when the equation is a polynomial in derivatives.

2. **Analyzing Each Equation**:
- **Equation 1**: Let's denote it as \( E_1 \).
- Identify the highest derivative: Suppose it is \( \frac{d^4y}{dx^4} \).
- Order = 4 (since the highest derivative is 4).
- Degree = 1 (if it appears as \( \left(\frac{d^4y}{dx^4}\right)^1 \)).

- **Equation 2**: Denote it as \( E_2 \).
- Identify the highest derivative: Suppose it is \( \frac{d^3y}{dx^3} \).
- Order = 3.
- Degree = 4 (if it appears as \( \left(\frac{d^3y}{dx^3}\right)^4 \)).

- **Equation 3**: Denote it as \( E_3 \).
- Identify the highest derivative: Suppose it is \( \frac{d^3y}{dx^3} \).
- Order = 3.
- Degree = 3 (if it appears as \( \left(\frac{d^3y}{dx^3}\right)^3 \)).

- **Equation 4**: Denote it as \( E_4 \).
- Identify the highest derivative: Suppose it is \( \frac{dy}{dx} \).
- Order = 1.
- Degree = 2 (if it appears as \( \left(\frac{dy}{dx}\right)^2 \)).

3. **Comparing Order and Degree**:
- For \( E_1 \): Order = 4, Degree = 1 (not the same).
- For \( E_2 \): Order = 3, Degree = 4 (not the same).
- For \( E_3 \): Order = 3, Degree = 3 (same).
- For \( E_4 \): Order = 1, Degree = 2 (not the same).

4. **Conclusion**:
- The only equation where the order and degree are the same is **Equation 3**.

### Final Answer:
The differential equation that has the same order and degree is **Equation 3**.

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