For the differential equation \( \displaystyle\frac{d}{dx}\left(\displaystyle\frac{d^3y}{dx^3}\right)+\left(\displaystyle\frac{dy}{dx}\right)^3+y+x=7 \), which of the following statements are True?
A. Order of differential equation is 4
B. Number of arbitrary constants in the particular solution of differential equation are 4
C. Number of arbitrary constants in the general solution of the differential equation are 4
D. Degree of differential equation is 3
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A and C only
Since \( \frac{d}{dx}\left(\frac{d^3y}{dx^3}\right)=\frac{d^4y}{dx^4} \), the equation is
\( \displaystyle \frac{d^4y}{dx^4}+\left(\frac{dy}{dx}\right)^3+y+x=7 \)
So the order is 4, the degree (power of \( \frac{d^4y}{dx^4} \)) is 1, the general solution has 4 arbitrary constants and a particular solution has none.
Checking the statements
(A) Order of differential equation is 4 — Correct. The highest order derivative is \( \frac{d^4y}{dx^4} \).
(B) Number of arbitrary constants in the particular solution of differential equation are 4 — Incorrect. A particular solution has no arbitrary constants.
(C) Number of arbitrary constants in the general solution of the differential equation are 4 — Correct. The general solution has as many arbitrary constants as the order, here 4.
(D) Degree of differential equation is 3 — Incorrect. The degree is 1, the power of \( \frac{d^4y}{dx^4} \); the power 3 belongs to \( \frac{dy}{dx} \).