Question

solve the initial value problem by the Laplace transform method: y''+4y=0 ; y(0)=1, y'(0)=1 solve the initial value problem by the Laplace transform method: y''+4y=0 ; y(0)=1, y'(0)=1 @Mathematics

Answer 1

\[y''+4y=0\quad;\quad y(0)=y'(0)=1\]\[s^2Y-sy(0)-y'(0)+4Y=0\]\[(s^2+4)Y=sy(0)+y'(0)=s+1\]\[Y=\frac{s+1}{s^2+4}=\frac{s}{s^2+4}+\frac{1}{s^2+4}=\frac{s}{s^2+2^2}+\frac{1}{2}\cdot\frac{2}{s^2+2^2}\]\[y=\cos{2t}+\frac{\sin{2t}}{2}\]

Answer 2

is there a way to check these through wolfram that you know of?

Answer 3

not your answer....my other work

Answer 4

http://www.wolframalpha.com/input/?i=y%27%27%2B4y%3D0+and+y%280%29%3D1+and+y%27%280%29%3D1