Michael J. answered • 05/10/16

Mastery of Limits, Derivatives, and Integration Techniques

First, rewrite the equation so that it looks like a first order differential equation.

x' = -3x + 300

x' + 3x = 300

Use the integrating factor:

u(t) = e

^{∫p(t)dt}where p(t)=3u(t) = e

^{∫3dt}u(t) = e

^{3t}Next, multiply both sides of the equation by the integrating factor.

e

^{3t}x' + 3xe^{3t}= 300e^{3t}Next, we integrate both sides of the equation. Notice that the left side is a product rule.

∫d/dt

**[**e^{3t}x**]**= 300∫e^{3t}e

^{3t}x = 300(1/3)e^{3t}+ Cwhere C is an arbitrary constant.

Divide both sides of the equation by e

^{3t}.x(t) = (300/3) + Ce

^{-3t}This is your general solution of the differential equation. Next, we need to solve for C using the initial value condition x(0)=80.

80 = (300/3) + Ce

^{-3(0)}80 = 300/3 + C

240 = 300 + 3C

-60 = 3C

-20 = C

x(t) = (300/3) - 20e

^{-3t}Now solve for t from this particular solution using the value x(t)=40.

40 = (300/3) - 20e

^{-3t}Take it from here.