Calculus: separable differential equations

[alexis36]

Homework Statement

The question is discussing the growth rate of a tumor as it decreases in size (called the Gompertz equation: I am needing so SOLVE THE SEPERABLE DIFFERENTIAL EQUATION.

Homework Equations

dx/dt = f(t)g(x) =(e^-t)(x(t)) x(o)=1

The Attempt at a Solution

1. I am separating variables, (i think I am making a mistake here)
dx/dt=(e^-t)x(t)
dx/x(t)=(e^-t)(dt)
.. is that step right?

2. then i integrate both sides.

integrating 1/x(t) on one side.. and integrating e^-t(dt) on the other.. i get
ln|x|= -e^-t + c
.. now I am unsure as to how to solve the rest. am i solving for a variable for c?

 

[HallsofIvy]

Homework Statement

The question is discussing the growth rate of a tumor as it decreases in size (called the Gompertz equation: I am needing so SOLVE THE SEPERABLE DIFFERENTIAL EQUATION.

Homework Equations

dx/dt = f(t)g(x) =(e^-t)(x(t)) x(o)=1

The Attempt at a Solution

1. I am separating variables, (i think I am making a mistake here)
dx/dt=(e^-t)x(t)
dx/x(t)=(e^-t)(dt)
.. is that step right?

2. then i integrate both sides.

integrating 1/x(t) on one side.. and integrating e^-t(dt) on the other.. i get
ln|x|= -e^-t + c
.. now I am unsure as to how to solve the rest. am i solving for a variable for c?

Looks to me like you have done everything right. Now use the fact that x(0)= 1 to find c. ln|1|= 0= -e^0+ c or 0= -1+ c. What is c? (c isn't, technically, a "variable", it is constant.)