Question

Show that y=cos(t)y=cos(t) is a solution to (dydt)2=1−y2(dydt)2=1−y2. Enter your answers below in terms of the independent variable tt in the order in which the terms were given. Be sure you can justify your answer.

Answer 1

Answer:

$(\frac{dy}{dt})^2=sin^2t$

Step-by-step explanation:

$y=cost$

DE :$(\frac{dy}{dt})^2=1-y^2$

If y is a solution of given DE then it satisfied the DE.

Differentiate w.r.t t

$\frac{dy}{dt}=-sint$

Using the formula

$\frac{d(cosx)}{dx}=-sinx$

LHS:$(\frac{dy}{dt})^2=(-sint)^2=sin^2t$

RHS

$1-y^2=1-cos^2t=sin^2t$

By using the formula

$sin^2t=1-cos^2t$

LHS=RHs

Hence, y is a solution of given DE

$(\frac{dy}{dt})^2=sin^2t$