Show that y=sin(t) is a solution to (dydt)2=1−y2. Enter your answers below in terms of the independent variable t in the order i
n which the terms were given. Be sure you can justify your answer.
1 answer:
Answer:
y = sin(t) is a solution to the differential equation
(dy/dt)² = 1 - y²
Step-by-step explanation:
Given (dy/dt)² = 1 - y²
Suppose y = sin(t) is a solution, then it satisfies the differential equation.
That is
[d(sin(t))]² = 1 - y²
Let y = sin(t)
dy/dt = d(sin(t)) = cos(t)
(dy/dt)² = cos²t
But cos²t + sin²t = 1
=> 1 - sin²t = cos²t
So
(dy/dt)² = 1 - sin²t
Since sin²t = (sint)² = y²,
we have
(dy/dt)² = 1 - y²
as required.
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