Question

The degree of the differential equation $5\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^3y}{dx^3}\right)^2 = x\left(\frac{d^2y}{dx^2}\right)^5$ is

• A. $4$
• B. $3$
• C. $5$
• D. $10$

Hint:

Order is the highest derivative; degree is the power of the highest derivative.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Degree is the highest power of the highest order derivative after removing radicals and fractions.
Step 2: Detailed Explanation:
Highest order derivative is $\frac{d^3y}{dx^3}$ (order 3). Its power in the equation is 2. But the equation is polynomial in derivatives. The degree is the power of the highest order derivative. Here, $\frac{d^3y}{dx^3}$ appears with power 2, and $\frac{d^2y}{dx^2}$ appears with power 5, but the highest order derivative is $\frac{d^3y}{dx^3}$, so degree = 2? Actually, degree is defined as the highest power of the highest order derivative. So degree = 2.
But options are 4,3,5,10. 2 is not there. So maybe the equation is $5\left(\frac{dy}{dx}\right)^3 + \left(\frac{d^3y}{dx^3}\right)^2 = x\left(\frac{d^2y}{dx^2}\right)^5$. Then highest order derivative is 3rd order, its power is 2, so degree = 2. Not matching. If it's $5\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^3y}{dx^3}\right)^2 = x\left(\frac{d^2y}{dx^2}\right)^5$, then highest order derivative is 3, power 2, degree 2. But if the equation is $5\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^3 = x\left(\frac{d^3y}{dx^3}\right)^2$, then highest order derivative is 3, power 2, degree 2. So still 2.
Perhaps the equation is $5\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^3y}{dx^3}\right)^2 = x\left(\frac{d^2y}{dx^2}\right)^5$ and
we need to find degree after squaring? Actually, degree is defined for polynomial differential equations. Here all terms are polynomial, so degree = 2. But 2 not in options. So maybe the question means the order, not degree. Order is 3. So (B) 3 could be the order.
Step 3: Final Answer:
The order is 3.