Question

solve by Laplace transform method
\[\Large y^{\prime \prime} + a^2 y =0; \quad y(0) = 0, \; y'(0) = a\]

Answer 1

my solution:
\[s^2 y(s) - sy(0) - y^\prime (0) + a^2 y(s) = 0\]
\[\implies s^2 y(s) - a + a^2 y(s) = 0\]
\[\implies y(s) (s^2 + a^2) - a = 0\]
\[\implies y(s) = \frac{a}{s^2 + a^2}\]
\[\implies y(t) = \sin at\]
am i right?

Answer 2

seems absolutely correct...

Answer 3

thanks for confirming