If \(y = y(x)\) is the solution of the differential equation \(\displaystyle\frac{d^2y}{dx^2} = 6x\) satisfying \(y(0) = 1\) and \(y'(0) = 2\), then \(y(1)\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 4
The equation has order 2, so its general solution has two arbitrary constants, and both conditions are needed.
Integrating once:
\( \displaystyle \frac{dy}{dx} = 3x^2 + C_1 \)
Putting x = 0, \(y'(0) = 2\):
\(C_1 = 2\)
Integrating again:
\(y = x^3 + 2x + C_2\)
Putting x = 0, y(0) = 1:
\(C_2 = 1\)
\(y = x^3 + 2x + 1\)
At x = 1:
\(y(1) = 1 + 2 + 1 = 4\)