Question

The general solution of \(\frac{d^2 y}{dx^2} = 0\) is

• A. General solution of \(\frac{d^3y}{dx^3} = 0\)
• B. General solution of \(\frac{dy}{dx} = a + b\)
• C. General solution for both \(\frac{d^2y}{dx^2}=0\) and \(\frac{d^3y}{dx^3}=0\)
• D. General solution for \(\frac{d^2y}{dx^2} = 0\)

Hint:

For a second-order differential equation of the form \(y'' = 0\), integrate twice to find general solution containing two arbitrary constants.

Answer 1

The Correct Option is D

Step 1: Identify the differential equation type.
Given \(\frac{d^2y}{dx^2} = 0\), it is a second-order linear differential equation with constant coefficients.

Step 2: Integrate first time.
\(\frac{dy}{dx} = C_1\) where \(C_1\) is constant of integration.

Step 3: Integrate second time.
\(y = C_1 x + C_2\), where \(C_2\) is another constant of integration.

Step 4: General solution form.
The solution contains two arbitrary constants \(C_1\) and \(C_2\), matching order of differential equation.

Step 5: Verify.
Differentiating \(y = C_1 x + C_2\) twice gives \(\frac{d^2y}{dx^2} = 0\) confirming correctness.

Step 6: Final conclusion.
Hence, the general solution is \[ \boxed{y = C_1 x + C_2} \]