Question

A particle moves along the x-axis with position at time t given by x(t)=e^(-t)sin(t) for 0 is less than or equal to t which is less than or equal to 2 pi.

a) Find the time t at which the particle is farthest to the left. Justify your answer

I think you have to find the prime of this equation and then see when it is negative.

b) Find the value of the constant A for which x(t) satisfies the equation Ax"(t)+x'(t)+x(t)=0 for 0 is less than t which is less than 2 pi.

I have no idea how to even start this problem.

Answer 1

a. Find the derivative, set to zero

dx/dt= -e^-t * sint+ e^-tcost=0

tanT=1 check that.

t= PI/4 or 3PI/4
Now which will make it to the left (negative x)?

b. d^2x/dt^2= d/dx e^-t(cost-sint)
take that dervative.

Then, put in the equation given
ax" + x'+ x=0 and solve for A