Question

Найти $frac{d^2y}{dx^2}$ функции

$left { {{x=3(t-sint)} atop {y=4(1-cost})} right.$

Answer 1

$x=phi (t);\ y=psi (t);\ y'_x=frac{psi'(t)}{phi'(t)};\ y''_{x^2}=frac{psi''(t)phi '(t)-psi'(t)phi''(t)}{(phi'(t))^3}$

 

$x'_t=3-3cost;\ x''_{t^2}=3sint;\ y'_t=4sint;\ y''_{t^2}=4cost;$

 

$y''_{x^2}=frac{4cost(3-3cost)-4sint *3sint}{(3-3cost)^3}=\ frac{12cost-12(cos^2 t+sin^2 t)}{27(1-cost)^3}=\ frac{12cost-12*1}{27(1-cost)^3}=\ frac{-12(1-cost)}{27(1-cost)^3}=\ frac{-4}{9(1-cost)^2}$