Question

If $x=a{e}^t(sint+cost)$ and $y=a{e}^t(sint-cost)$, then find$\frac{dx}{dy}$.

Answer 1

$x=a{e}^t(\sin t+\cos t),y=a{e}^t(\sin t-\cos t)$

$\frac{dx}{dt}=a{e}^t\sin t+a{e}^t\cos t+a{e}^t\cos t-a{e}^t\sin t$

$\frac{dx}{dt}=2a{e}^t\cos t$

$\frac{dy}{dx}=a{e}^t\sin t+a{e}^t/\cos t-a{e}^t/\cos t+a{e}^t\sin t$

$\frac{dy}{dt}=2a{e}^t\sin t$

$\therefore \frac{dx}{dy}=\frac{\frac{dx}{dy}}{\frac{dy}{dt}}=\frac{2a{e}^t\cos t}{2a{e}^t\sin t}=\frac{\cos t}{\sin t}=\cot t$.