Solve y'' + y = 0 with y(0) = 2 and y'(0) = 0. Which function satisfies the equation?

Boost your skills in A Level Further Mathematics Core Pure. Study with flashcards and multiple choice questions, each question is followed by hints and explanations. Get prepared for your exam with confidence!

This differential equation describes simple harmonic motion, and its general solution is a linear combination of sine and cosine: y = A cos x + B sin x. Use the initial conditions to find A and B. At x = 0, y(0) = A = 2, so A = 2. The derivative is y' = -A sin x + B cos x, and y'(0) = B = 0, so B = 0. Therefore the function is y = 2 cos x, which indeed satisfies y'' + y = 0 since y'' = -2 cos x and -2 cos x + 2 cos x = 0. The other forms fail the initial conditions: 2 sin x gives y(0) = 0, cos x gives y(0) = 1, and the constant 2 does not satisfy the differential equation.