If \( m \) and \( n \) are respectively order and degree of the differential equation
\( \left(\displaystyle\frac{d^2y}{dx^2}\right)^5+6.\displaystyle\frac{\left(\displaystyle\frac{d^2y}{dx^2}\right)^3}{\displaystyle\frac{d^3y}{dx^3}}+\displaystyle\frac{d^3y}{dx^3}=x^2+5 \) then :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → m = 3, n = 2
The highest order derivative present is \( \frac{d^3y}{dx^3} \), so the order is m = 3.
The degree can be read only when the equation is a polynomial in its derivatives. Here \( \frac{d^3y}{dx^3} \) sits in a denominator, so the whole equation is multiplied by \( \frac{d^3y}{dx^3} \):
\( \displaystyle \left(\frac{d^2y}{dx^2}\right)^5\frac{d^3y}{dx^3}+6\left(\frac{d^2y}{dx^2}\right)^3+\left(\frac{d^3y}{dx^3}\right)^2=(x^2+5)\frac{d^3y}{dx^3} \)
The highest power of the highest order derivative \( \frac{d^3y}{dx^3} \) is now 2, so n = 2.
Hence m = 3, n = 2.
Why the other options are incorrect
Option (1) m = 3, n = 3 — Incorrect. 3 is the power of \( \frac{d^2y}{dx^2} \) in the middle term; the degree is fixed by the highest order derivative, \( \frac{d^3y}{dx^3} \), whose highest power is 2.
Option (2) m = 2, n = 3 — Incorrect. The order is 3, not 2, because \( \frac{d^3y}{dx^3} \) is present.
Option (4) m = 3, n = 5 — Incorrect. 5 is the power of \( \frac{d^2y}{dx^2} \), which is not the highest order derivative; the fraction must also be cleared first.