Question

Ap calc please help (question 4)

Answer 1

a)


`y(t)=5\sin(t)-\cos(t)\\\\y'(t)=5\cos(t)+\sin(t)\\\\y''(t)=-5\sin(t)+\cos(t)\\\\\\7y(t)-3y''(t)=7(5\sin(t)-\cos(t))-3(-5\sin(t)+\cos(t))=\\\\=35\sin(t)-7\cos(t)+15\sin(t)-3\cos(t)=\\\\=50\sin(t)-10\cos(t)=5\big(10\sin(t)-2\cos(t)\big)\\eq5\cos(t)`

Not a solution.

b)


`y(t)=\sin(t)\\\\y'(t)=\cos(t)\\\\y''(t)=-\sin(t)\\\\\\7y(t)-3y''(t)=7\sin(t)+3\sin(t)=10\sin(t)\\eq5\cos(t)`

Not a solution.

c)


`y(t)=(1)/(2)\sin(t)\\\\y'(t)=(1)/(2)\cos(t)\\\\y''(t)=-(1)/(2)\sin(t)\\\\\\7y(t)-3y''(t)=(7)/(2)\sin(t)+(3)/(2)\sin(t)=(10)/(2)\sin(t)=5\sin(t)\\eq5\cos(t)`

Not a solution.

d)


`y(t)=(1)/(2)\cos(t)\\\\y'(t)=-(1)/(2)\sin(t)\\\\y''(t)=-(1)/(2)\cos(t)\\\\\\7y(t)-3y''(t)=(7)/(2)\cos(t)+(3)/(2)\cos(t)=(10)/(2)\cos(t)=\boxed{5\cos(t)\quad\checkmark}`

e)


`y(t)=\cos(t)\\\\y'(t)=-\sin(t)\\\\y''(t)=-\cos(t)\\\\\\7y(t)-3y''(t)=7\cos(t)+3\cos(t)=10\cos(t)\\eq5\cos(t)`

Not a solution.

Correct answer is d)