Question

For the differential equation \[ \frac{d^3y}{dx^3}=0, \] \(y=ax^2+bx+c\) is

• A. the general solution
• B. a particular solution
• C. not a solution
• D. a solution, but not a particular solution

Hint:

For an \(n\)-th order differential equation, the general solution contains \(n\) arbitrary constants.

Answer 1

The Correct Option is A

Step 1: Integrate the ODE.
\(\frac{d^3y}{dx^3}=0\). Integrate once: \(\frac{d^2y}{dx^2}=A\). Integrate again: \(\frac{dy}{dx}=Ax+B\). Integrate once more: \(y=\frac{A}{2}x^2+Bx+C\).

Step 2: Match with y=ax2+bx+c.
Setting \(a=\frac{A}{2},b=B,c=C\), this is exactly \(y=ax^2+bx+c\) with 3 arbitrary constants \(a,b,c\). Since the ODE is 3rd order, this 3-parameter family is the general solution. \[\boxed{\text{the general solution}}\]