Choose the correct statements:
A. The order and degree (if defined) of a differential equation are always positive integers
B. The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation
C. If \(\displaystyle\frac{dy}{dx} + P(x)\,y = Q(x)\) then Integrating factor \(= e^{\int P(x)dx}\)
D. The sum of order and degree of differential equation \(1 + (y'')^5 = (y''')^3\) is 8
E. If the solution of a differential equation of order n, contains n arbitrary constants, then it is called a general solution
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → A, B, C, E only
In \(1 + (y'')^5 = (y''')^3\) the highest derivative is y''', so the order is 3. The equation is a polynomial in the derivatives and y''' has power 3, so the degree is 3.
\(\text{order} + \text{degree} = 3 + 3 = 6\)
So order and degree are always positive integers, the integrating factor of \(\frac{dy}{dx} + P(x)y = Q(x)\) is \(e^{\int P(x)dx}\), and for \(1 + (y'')^5 = (y''')^3\) the sum of order and degree is 6.
Checking the statements
(A) The order and degree (if defined) of a differential equation are always positive integers — Correct. The order counts derivatives and the degree is a power, both positive whole numbers.
(B) The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable involved in a differential equation — Correct. This is the definition of order.
(C) If \(\frac{dy}{dx} + P(x)\,y = Q(x)\) then Integrating factor \(= e^{\int P(x)dx}\) — Correct. This is the integrating factor of a linear equation.
(D) The sum of order and degree of differential equation \(1 + (y'')^5 = (y''')^3\) is 8 — Incorrect. Order 3 + degree 3 = 6, not 8.
(E) If the solution of a differential equation of order n, contains n arbitrary constants, then it is called a general solution — Correct. This is the definition of the general solution.