Question

Find dy/dx and d²y/dx² without eliminating the parameter.

a.) x= t^2 e^t , y= t In(t)

Answer 1

$\dot x=\frac{d}{dt}(t^2e^t) = ( t^2 +2t)e^t$

$\dot y=\frac{d}{dt}(t\ln t) = \ln t + 1$

$\frac{dy}{dx} = \frac{\dot y}{\dot x} = \frac{\ln t +1}{(t^2+2t)e^t}$

$\frac{d}{dt}\frac{dy}{dx} = \frac{d}{dt}\frac{\ln t +1}{(t^2+2t)e^t} = \frac{\frac{1}{t}(t^2+2t)e^t - (\ln t +1)e^t(t^2+2t + 2t+2)}{(e^t(t^2+2t))^2} = \frac{t+2-(\ln t+1)(t^2+4t+2)}{(t^2+2t)^2}e^{-t}$

$\frac{d^2y}{dx^2} = \frac{1}{\dot x}\frac{d}{dt}\frac{dy}{dx} = \frac{t+2-(\ln t+1)(t^2+4t+2)}{(t^2+2t)^3}e^{-2t}$