For what value of t is x(t)=40?

Question

Solve the equation x′=3(100−x) with the initial condition x(0)=80.

 

 For what value of t is x(t)=40?

 

(university level math)

Answer 1

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)=3

 

u(t) = e^∫3dt

 

u(t) = e^3t

 

 

Next, multiply both sides of the equation by the integrating factor.

 

e^3tx' + 3xe^3t = 300e^3t

 

 

Next, we integrate both sides of the equation.  Notice that the left side is a product rule.

 

∫d/dt[e^3tx] = 300∫e^3t

 

 

e^3tx = 300(1/3)e^3t + C

 

where 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.