(d^3y/dx)^4/3 +(dy/dx)^1/5 +y =0 find order and degree

To find the order and degree of the differential equation:

\[
\left(\frac{d^3y}{dx^3}\right)^{\frac{4}{3}} + \left(\frac{dy}{dx}\right)^{\frac{1}{5}} + y = 0
\]

### Order:
The **order** of a differential equation is the highest derivative of the dependent variable (here, \(y\)) with respect to the independent variable (here, \(x\)).

- The highest derivative in the equation is \(\frac{d^3y}{dx^3}\), which is the third derivative.

So, the **order** of the differential equation is **3**.

### Degree:
The **degree** of a differential equation is the power of the highest order derivative, provided the equation is a polynomial in derivatives.

- However, in this case, the derivative \(\frac{d^3y}{dx^3}\) is raised to the power of \(\frac{4}{3}\) and \(\frac{dy}{dx}\) is raised to the power of \(\frac{1}{5}\). Both of these are fractional powers, meaning the equation is not a polynomial in the derivatives.

Thus, the differential equation **does not have a degree**.

### Summary:
- **Order**: 3
- **Degree**: Not defined (because of the fractional powers).

Would you like more details or have any other questions? 

Here are five related questions you might be interested in:
1. What is the significance of the order of a differential equation?
2. How do you find the degree if the equation was polynomial in derivatives?
3. Can you provide an example of a differential equation with a defined degree?
4. What is the difference between linear and non-linear differential equations?
5. How does the fractional power affect the nature of the differential equation?

**Tip:** When determining the degree of a differential equation, always ensure it is in polynomial form with respect to the highest derivative before identifying the degree.

Explore how to determine the order and degree of a differential equation involving fractional powers of derivatives. This problem analysis covers the highest derivative, order determination, and the absence of a degree due to fractional powers. Ideal for advanced college-level students and those studying differential equations.

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