Question

Let \( c_1, c_2, c_3, c_4 \) be arbitrary constants. The order of the differential equation corresponding to \[ y = c_1 e^x + c_2 e^{\log_e x} + c_3 \sin^2 x - c_4 (\cos^5 x - 1) \] is:

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Order of a differential equation is determined by the highest derivative needed to eliminate all arbitrary constants.

Answer 1

The Correct Option is C

The function \( y \) is a combination of exponential, logarithmic, and trigonometric terms.
- \( e^x \) and \( e^{\log x} = x \) are of order 1.
- \( \sin^2 x \) differentiates to terms involving \( \sin x \cos x \) and continues.
- \( \cos^5 x - 1 \) involves higher-order trigonometric expressions.
To remove all arbitrary constants, the maximum derivative required would be 3rd derivative.