Question

d(xt)/dt = x .plz explain.... best answer is marked branliest

Answer 1

Given x = a(t - sint) = (at - asint)
⇒ dx/dt = a - acost = a(1 - cost)
Given y = a(1 + cost) = a + acost
⇒ dy/dt = -asint
⇒ (dy/dt)/(dx/dt) = -asint/a(1 - cost)
⇒ dy/dx = sint/(cost - 1)
we know that sint = 2sin(t/2).cos(t/2) and cost - 1 = -2sin²(t/2)
substituting we get :
⇒ dy/dx = 2sin(t/2).cos(t/2)/-2sin²(t/2) = -cos(t/2)/sin(t/2) = -cot(t/2)
⇒ dy/dx = -cot(t/2)
⇒ d²y/dx² = d/dx × [-cot(t/2)]
⇒ d²y/dx² = -(-cosec²(t/2) × 1/2 × dt/dx)
we know that dx/dt = a - acost = a(1 - cost)
⇒ dt/dx = 1/a(1 - cost)
⇒ d²y/dx² = 1/2 × cosec²(t/2) × 1/a(1 - cost)
we know that (1 - cost) = 2sin²(t/2)
⇒ d²y/dx² = 1/2a × cosec²(t/2) × 1/2sin²(t/2)
⇒ d²y/dx² = 1/2a × cosec²(t/2) × cosec²(t/2) × 1/2
⇒ d²y/dx² = 1/4a × cosec⁴(t/2)

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