The order and the degree of the differential equation \(\displaystyle\frac{d^3y}{dx^3} + x\sin\left(\frac{d^2y}{dx^2}\right) = y\) are
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → order 3, degree not defined
The highest derivative present is \(\frac{d^3y}{dx^3}\), so the order is 3.
The degree is defined only when the equation is a polynomial in its derivatives. Here the derivative \(\frac{d^2y}{dx^2}\) appears inside the sine:
\( \displaystyle \sin\left(\frac{d^2y}{dx^2}\right) = \frac{d^2y}{dx^2} - \frac{1}{3!}\left(\frac{d^2y}{dx^2}\right)^3 + \cdots \)
This is not a polynomial in \(\frac{d^2y}{dx^2}\), so the degree is not defined, even though \(\frac{d^3y}{dx^3}\) itself has power 1.
Why the other options are incorrect
Option (2) order 3, degree 1 — Incorrect. The power 1 of \(\frac{d^3y}{dx^3}\) cannot be taken as the degree, because the term \(\sin\left(\frac{d^2y}{dx^2}\right)\) makes the equation non-polynomial in the derivatives.
Option (3) order 2, degree not defined — Incorrect. The order is set by the highest derivative \(\frac{d^3y}{dx^3}\), which is 3, not by the derivative inside the sine.
Option (4) order 2, degree 1 — Incorrect. The third derivative is present, so the order is 3, and the degree is not defined.