To determine the order and degree of a differential equation, we analyze the highest order derivative and its degree, keeping in mind that the equation needs to be free from fractions and radicals of derivatives for the degree determination.
Let's solve each equation step by step:
1. Equation: [tex]\(\frac{d^4 y}{dx^4} + \sin(y''') = 0\)[/tex]
- Order: The highest derivative present is [tex]\(\frac{d^4 y}{dx^4}\)[/tex], so the order is 4.
- Degree: Since this equation has [tex]\(\sin(y''')\)[/tex], the highest power of the highest order derivative (which is linear) indicates that the degree is 1.
Therefore, Order: 4, Degree: 1.
2. Equation: [tex]\(y' + 5y = 0\)[/tex]
- Order: The highest derivative present is [tex]\(y'\)[/tex] (the first derivative), so the order is 1.
- Degree: The highest power of [tex]\(y'\)[/tex] is 1 (implied as there are no powers written explicitly).
Therefore, Order: 1, Degree: 1.
3. Equation: [tex]\(\left(\frac{d s}{d t}\right)^4 + 3s\left(\frac{d^2 5}{dt^2}\right) = 0\)[/tex]
- Order: The highest derivative present is [tex]\(\frac{d^2 s}{dt^2}\)[/tex], so the order is 2.
- Degree: Since the equation contains [tex]\(\left(\frac{d s}{d t}\right)^4\)[/tex], the highest power of the first-order derivative [tex]\(\frac{d s}{dt}\)[/tex] indicates that the degree is 4.
Therefore, Order: 2, Degree: 4.
4. Equation: [tex]\(\left(\frac{d^2 y}{dx^2}\right)^2 + \cos\left(\frac{d y}{dx}\right) = 0\)[/tex]
- Order: The highest derivative present is [tex]\(\frac{d^2 y}{dx^2}\)[/tex], so the order is 2.
- Degree: The highest power of the highest order derivative included in the equation, which is [tex]\(\left(\frac{d^2 y}{dx^2}\right)^2\)[/tex], indicates that the degree is 2.
Therefore, Order: 2, Degree: 2.
5. Equation: [tex]\(\frac{d^2 y}{dx^2} = \cos(3x) + \sin(x)\)[/tex]
- Order: The highest derivative present is [tex]\(\frac{d^2 y}{dx^2}\)[/tex], so the order is 2.
- Degree: Since the highest power of [tex]\(\frac{d^2 y}{dx^2}\)[/tex] is 1 (implied as there are no powers written explicitly), the degree is 1.
Therefore, Order: 2, Degree: 1.
Summarizing, the order and degree of the given differential equations are:
[tex]\[
\begin{align*}
1. & \quad \text{Order: 4, Degree: 1} \\
2. & \quad \text{Order: 1, Degree: 1} \\
3. & \quad \text{Order: 2, Degree: 4} \\
4. & \quad \text{Order: 2, Degree: 2} \\
5. & \quad \text{Order: 2, Degree: 1}
\end{align*}
\][/tex]
