How Is the Tumor Growth Equation Solved Using Separation of Variables?

[ninaricci]

can anyone show me how to solve this equation that represents the tumor growth
ds/dt = - a s ln(bs)
where a and b are constants

 

[Tide]

First, change variable to u = bs which gives

$$\frac {du}{dt} = -a u \ln u$$

which you can integrate by setting

$$\frac {du}{u \ln u} = -a dt$$

so that

$$\ln \frac {\ln u}{\ln u_0} = -a t$$

where $u_0$ is the initial condition. You can exponentiate both sides twice to obtain an explicit solution for u(t) and finally s(t).

 

[M98Ranger]

To solve the tumor growth equation, we can use the separation of variables method. This involves separating the variables on each side of the equation and integrating both sides.

First, we can rewrite the equation as:

ds/ln(bs) = -a dt

Next, we can integrate both sides of the equation. On the left side, we can use the substitution u = ln(bs) and du = (1/s) ds to get:

∫1/s ds = ∫-a dt

ln(s) = -at + C

Where C is the constant of integration.

To solve for s, we can take the exponential of both sides:

s = e^(-at + C)

Using the properties of logarithms, we can rewrite this as:

s = e^C * e^(-at)

Since e^C is just a constant, we can rewrite it as another constant, let's call it K.

s = Ke^(-at)

Therefore, the solution to the tumor growth equation is:

s(t) = Ke^(-at)

Where K is a constant determined by the initial conditions of the tumor. This solution shows that the tumor growth decreases exponentially over time, with the rate of decrease determined by the values of a and b.