Question

Among the options given below, from which option a differential equation of order two can be formed?

• A. All circles passing through the origin
• B. All parabolas passing through the origin and having focus on x-axis
• C. All the lines passing through the origin
• D. All hyperbolas of the form \( x^2 - y^2 = k^2 \)

Hint:

The order of the differential equation is equal to the number of arbitrary constants in the general solution.

Answer 1

The Correct Option is A

(1) All circles passing through the origin. The general equation of a circle passing through the origin is \[ (x-a)^2 + (y-b)^2 = a^2 + b^2 \] \[ x^2 - 2ax + a^2 + y^2 - 2by + b^2 = a^2 + b^2 \] \[ x^2 - 2ax + y^2 - 2by = 0 \] \[ x^2 + y^2 = 2ax + 2by \] There are two arbitrary constants \(a\) and \(b\), so we can form a differential equation of order two. 
(2) All parabolas passing through the origin and having focus on x-axis. The equation of such parabolas is \( y^2 = 4ax \). There is only one arbitrary constant \(a\), so we can form a differential equation of order one. 
(3) All the lines passing through the origin. The equation of such lines is \( y = mx \). There is only one arbitrary constant \(m\), so we can form a differential equation of order one. 
(4) All hyperbolas of the form \( x^2 - y^2 = k^2 \). There is only one arbitrary constant \(k\), so we can form a differential equation of order one. Thus, the only option that can form a differential equation of order two is (1).