The differential equation provided is:

$2x^4 \frac{d^4 y}{dx^4} + 5x^2 \left( \frac{dy}{dx} \right)^3 - xy = 0$

To determine the order and degree of this differential equation:

1. Order:

• The order of a differential equation is the highest derivative present in the equation.
• In this case, the highest derivative is $\frac{d^4 y}{dx^4}$, which is the fourth derivative of $y$.
• Therefore, the order of the differential equation is 4.

2. Degree:

• The degree of a differential equation is the power of the highest order derivative, provided the equation is polynomial in the derivatives.
• In the given equation, the term $\frac{d^4 y}{dx^4}$ appears linearly (its power is 1), and all other derivatives, such as $\left( \frac{dy}{dx} \right)^3$, are also raised to integer powers.
• Therefore, the degree of the differential equation is 1.

Conclusion:

• The order is 4 and the degree is 1, which corresponds to option (a) 4, 1.

Would you like further details or clarifications?

Related Questions:

1. What is the general form of a linear differential equation?
2. How do you solve a fourth-order linear differential equation?
3. What are the conditions for a differential equation to be considered exact?
4. Can the degree of a differential equation be non-integer?
5. What is the difference between partial and ordinary differential equations?

Tip: Always check if a differential equation is polynomial in its derivatives before determining the degree.