it has degree = 60 ,is there any way to find that

Yes, the differential equation can have a defined degree if we first eliminate the fractional powers by raising both sides of the equation to appropriate powers to make the exponents integers.

Let's rewrite the given differential equation:

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

To eliminate the fractional powers, follow these steps:

### Step 1: Eliminate the Fractional Powers
- Raise the entire equation to the power of 3 to eliminate the fractional power of \(\frac{4}{3}\) from \(\left(\frac{d^3y}{dx^3}\right)\):
  
\[
\left[\left(\frac{d^3y}{dx^3}\right)^{\frac{4}{3}} + \left(\frac{dy}{dx}\right)^{\frac{1}{5}} + y\right]^3 = 0^3
\]

Simplifying:

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

### Step 2: Further Eliminate the Fractional Power
- Now, raise the entire equation to the power of 5 to eliminate the fractional power of \(\frac{3}{5}\) from \(\left(\frac{dy}{dx}\right)\):

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

Simplifying:

\[
\left(\frac{d^3y}{dx^3}\right)^{20} + \left(\frac{dy}{dx}\right)^3 + y^{15} = 0
\]

### Step 3: Identify the Degree
- Now the highest derivative \(\frac{d^3y}{dx^3}\) is raised to the power of 20, and the equation is a polynomial in the derivatives.

So the **degree** of the differential equation is **20**.

**Correction**: You initially mentioned the degree as 60, which would imply a step might have raised the equation further. But based on these steps, the degree should be **20**. If the degree of 60 is somehow correct (maybe considering additional steps or mistakes in earlier assumptions), the process outlined would still be generally how you address fractional powers, but for the provided equation, it would be **20**.

Would you like to explore this further or have any other questions?

Here are five related questions you might be interested in:
1. Can you show an example where the degree of a differential equation is exactly 60?
2. What happens if we have other fractional powers in a differential equation?
3. How do fractional powers affect the solutions of differential equations?
4. What are the implications of having a high degree in a differential equation?
5. How do we solve a non-linear differential equation like this one?

**Tip:** Always check for hidden assumptions or factors that might alter the calculation, especially when dealing with fractional exponents in differential equations.

Explore how to determine the degree of a differential equation involving fractional powers. Learn step-by-step methods to eliminate fractional exponents and identify the correct degree. This problem involves advanced mathematical concepts in differential equations.

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