How to Find S as a Function of t in Differential Equations?

  • Context: Undergrad 
  • Thread starter Thread starter olechka722
  • Start date Start date
  • Tags Tags
    Integration Model
Click For Summary
SUMMARY

The discussion focuses on solving the differential equation dS/dt = A - (B*S/(S+C))*(D-E*t) to find S as a function of t. The participants clarify that direct integration is not feasible due to the presence of S on both sides of the equation. While the equation can be simplified if A equals 0, making it separable, the general case requires alternative methods. Ultimately, numerical integration is suggested as a viable approach for finding S.

PREREQUISITES
  • Understanding of first-order differential equations
  • Familiarity with numerical integration techniques
  • Knowledge of variable separation in differential equations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Explore numerical integration methods such as the Runge-Kutta method
  • Study separable differential equations and their solutions
  • Learn about the existence and uniqueness theorem for differential equations
  • Investigate software tools for solving differential equations, such as MATLAB or Python's SciPy library
USEFUL FOR

Mathematicians, engineering students, and anyone involved in solving differential equations or applying numerical methods in their work.

olechka722
Messages
6
Reaction score
0
Hi, it has been a little while since I have actually had a course in this, and my math memory is terrible. I would like to find S as a function of t, when the differential relationship is the following:

dS/dt= A- (B*S/(S+C))*(D-E*t) where all of the other letters are just constants.

What would be the correct approach here? Is it just numerical integration?
 
Physics news on Phys.org
If you mean just integrate both sides, no, it's not. You are looking for the unknown function S and you have S on the right side as well so you can't just integrate with respect to t.

That is, rather a differential equation.

If A were equal to 0, it would be "separable". We could "separate" the variables S and t as
\frac{S+C}{S}\frac{dS}{dt}= -B(D- Et)
and you can integrate both sides of that with respect to s:
\int\frac{S+C}{S}\frac{dS}{dt}dt= \int(-BD+ BEt)dt
\int\frac{S+C}{S}dS= \int(-BD+ BEt)dt.

However, with that "A", it's not that simple. There are a variety of ways of solving first order differential equation but I don't see any immediately that would work for that equation.
 
That is pretty much what I thought. Thank you!

I am going to just do numerical integration.
 

Similar threads

Graduate Memory issues in numerical integration of oscillatory function
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Multivariable fundamental calculus theorem in Wald
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Landscape Fabric Roll
  • · Replies 8 ·
Replies
8
Views
3K
Undergrad Find the value of this definite integral in terms of t, s and alpha
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Gaussian integral by differentiating under the integral sign
  • · Replies 19 ·
Replies
19
Views
5K
Undergrad Am I using the right limits on this triple integral?
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Evaluating Accuracy of Dynamic Model for Spreading Disease
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Integral representation of incomplete gamma function
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Ambiguity of the term "indefinite integral"
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Lipschitz continuity of vector-valued function
  • · Replies 3 ·
Replies
3
Views
4K