For the differential equation \( \displaystyle\frac{\left[1+\left(\frac{dy}{dx}\right)^{2}\right]^{\frac{3}{2}}}{\frac{d^{2}y}{dx^{2}}}=k,\ k\ne0 \), which one of the following is TRUE?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Non-linear differential equation of order 2 and degree 2
The degree is read only after the equation is free of fractional powers of the derivatives. Clearing the fraction,
\( \displaystyle \left[1+\left(\frac{dy}{dx}\right)^{2}\right]^{\frac{3}{2}}=k\,\frac{d^{2}y}{dx^{2}} \)
Squaring both sides,
\( \displaystyle \left[1+\left(\frac{dy}{dx}\right)^{2}\right]^{3}=k^{2}\left(\frac{d^{2}y}{dx^{2}}\right)^{2} \)
The highest derivative is \( \frac{d^{2}y}{dx^{2}} \), so the order is 2. It appears with power 2, so the degree is 2.
The derivatives appear with powers higher than 1, so the equation is non-linear.
Why the other options are incorrect
Option (2) linear differential equation of order 2 and degree 2 — Incorrect. The derivatives appear with powers higher than 1, so the equation is not linear.
Option (3) Non-linear differential equation of order 2 and degree 3 — Incorrect. 3 is the power of the bracket containing \( \frac{dy}{dx} \); the degree is the power of the highest derivative \( \frac{d^{2}y}{dx^{2}} \), which is 2.
Option (4) linear differential equation of order 2 and degree 3 — Incorrect. The equation is non-linear, and its degree is 2.