If m and n are the order and the degree of the differential equation :
\( x\displaystyle\frac{d^2y}{dx^2}=\left[1+\left(\frac{dy}{dx}\right)^2\right]^{1/3} \) then which one of the following is TRUE?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 3m + n = 9
The highest derivative is \( \frac{d^2y}{dx^2} \), so the order is m = 2.
The degree is defined only when the equation is a polynomial in the derivatives, so the fractional power is removed by cubing both sides:
\( \displaystyle x^3\left(\frac{d^2y}{dx^2}\right)^3=1+\left(\frac{dy}{dx}\right)^2 \)
Now the highest derivative \( \frac{d^2y}{dx^2} \) occurs to the power 3, so the degree is n = 3.
\( \displaystyle 3m+n=3(2)+3=9 \)
The other relations fail: m ≠ n; m + 2n = 8, not 7; 2m + n = 7, not 8.