Answer to Question #155305 in Calculus for Pia

Question

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

a.) x= e^(2t) , y= 1+cos(t)

b.) x= acosh(t) , y= bsinh(t)

Answer 1

Solution

a.)   x= e^(2t) , y= 1+cos(t) =>

dy/dx=(dy/dt)/(dx/dt)= (-sin(t))/(2e^2t)=-e^-2t*sin(t)/2

d²y/dx²=(d(dy/dx)/dt)/(dx/dt)=(e^-2t *sin(t)- e^-2t *cos(t)/2)/(2e^2t)= e^-4t (2*sin(t)-cos(t))/4 

b.)   x= acosh(t) , y= bsinh(t) =>

dy/dx=(dy/dt)/(dx/dt)=(b/a)*cosh(t)/sinh(t)=(b/a)*coth(t)

d²y/dx²=(d(dy/dx)/dt)/(dx/dt)=(b/a)*(-cosech(t)²)/(a*sinh(t)) = -b/(a²*sinh(t)³)

 Answer

a.)  dy/dx=-e^-2t*sin(t)/2, d²y/dx²=e^-4t (2*sin(t)-cos(t))/4 

b.)  dy/dx=(b/a)*coth(t), d²y/dx²= -b/(a²*sinh(t)³)