Question

1. (a) if f(t)=e^-wt sin wt, show that d^2f/dt^2 +2w df/dt + 5wf=0

(b) A curve passes through the points (1, 2) and its gradient function is 3x^4-1 all over X^2. Find the equation of the curve.

Answer 1

(a)

$$f'(t)=-we^{-wt}\sin wt+we^{-wt}\cos wt$$

$$f''(t)=w^2e^{-wt}\sin wt-w^2e^{-wt}\cos wt$$

$$-w^2e^{-wt}\cos wt-w^2e^{-wt}\sin wt$$

$$=-2w^2e^{-wt}\cos wt$$

$$\dfrac{d^2f}{dt^2}+2w\dfrac{df}{dt}+5wf$$

$$=-2w^2e^{-wt}\cos wt+2w(-we^{-wt}\sin wt+we^{-wt}\cos wt)$$

$$+5w(e^{-wt}\sin wt)=-2w^2e^{-wt}\sin wt+5we^{-wt}\sin wt$$

$$=-2w^2e^{-wt}\sin wt+5we^{-wt}\sin wt\not=0$$

(b)

$$f(x)=\int(3x^4-1)dx=\dfrac{3}{5}x^5-x+C$$

$$f(1)=\dfrac{3}{5}(1)^5-1+C=2=>C=\dfrac{12}{5}$$

$$f(x)=\dfrac{3}{5}x^5-x+\dfrac{12}{5}$$