The order and degree of differential equation \( y=px+\sqrt{a^2p^2+b^2},p=\displaystyle\frac{dy}{dx} \), \( a \) and \( b \) are constants are:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → order = 1, degree = 2
Only the first derivative p appears, so the order is 1. The degree is read only after the equation is free of the square root:
\( \displaystyle y-px=\sqrt{a^2p^2+b^2} \)
\( (y-px)^2=a^2p^2+b^2 \)
The highest power of p is now 2, so the degree is 2.
Why the other options are incorrect
Option (1) order = 1, degree = 1 — Incorrect. This reads the degree before the square root is removed. After squaring, p appears to the power 2.
Option (2) order = 2, degree = 1/2 — Incorrect. Only the first derivative appears, so the order is 1, and a degree is never a fraction.
Option (3) order = 1, degree not defined — Incorrect. The degree is not defined only when the derivative appears inside a function such as sin or log. A square root can be removed by squaring, so the degree is defined here.