# What kind of differential equation is this

I am trying to solve a real-world problem, and I have modeled it with the following equation:
$$\frac{dS}{dt}-3\left (\frac{150-S}{100-t} \right )+\frac{2S}{100+t}=0$$

What kind of differential equation is that? Is it solvable analytically ?
Maple tells me it is linear, but I don't see how...

Last edited:

lurflurf
Homework Helper
I forget what that is called, but it is linear and easily solved. Clear denominators
(t-100)(t+100)s'(t) = (t+500) s(t)-450 t-45000
solve
(t-100)(t+100) s'(t) = (t+500) s(t)
use that to solve the original
naturally terms like
(t-100)m(t+100)n
arise

Last edited:
I am trying to solve a real-world problem, and I have modeled it with the following equation:
$$\frac{dS}{dt}-3\left (\frac{150-S}{100-t} \right )+\frac{2S}{100+t}=0$$

What kind of differential equation is that? Is it solvable analytically ?
Maple tells me it is linear, but I don't see how...

It is a first order linear ODE
Of course it is easy to solve it, thanks to classical method.
"Linear" means linear relatively to the sought function S (There is no S² nor other functions of S. There are only S and S' in the equation).
It doesn't mean linear relatively to the variable t.

Last edited:
Write: dS/dt + f(t)S = g(t).

I am trying to solve a real-world problem, and I have modeled it with the following equation:
$$\frac{dS}{dt}-3\left (\frac{150-S}{100-t} \right )+\frac{2S}{100+t}=0$$

What kind of differential equation is that? Is it solvable analytically ?
Maple tells me it is linear, but I don't see how...

Looks like rate of equation that determines solution or volume of liquid into a tank and out. Again it is solvable via separable equations.