Applications in Physics and Modeling — Theory
1. Why differential equations are used in modeling
A differential equation describes how a quantity changes. In physics and real-world modeling,
the unknown function might represent position, charge, population, or amount of solute.
\[
\text{change rate}=\text{rule depending on the current state}
\]
This calculator focuses on four common application families:
- spring-mass systems,
- series RLC circuits,
- population growth,
- mixing tank problems.
2. Damped spring-mass model
A mass attached to a spring with damping is modeled by:
\[
m x^{\prime\prime}+c x^\prime+kx=0
\]
Here:
- \(m\) is the mass,
- \(c\) is the damping coefficient,
- \(k\) is the spring constant,
- \(x(t)\) is displacement from equilibrium.
Dividing by \(m\) gives:
\[
x^{\prime\prime}+\frac{c}{m}x^\prime+\frac{k}{m}x=0
\]
3. Spring-mass parameters
It is useful to define:
\[
\gamma=\frac{c}{2m},
\qquad
\omega_0=\sqrt{\frac{k}{m}}
\]
The equation becomes:
\[
x^{\prime\prime}+2\gamma x^\prime+\omega_0^2x=0
\]
The value of \(\gamma\) compared with \(\omega_0\) determines the type of motion.
5. Underdamped oscillator solution
If \(\gamma<\omega_0\), define:
\[
\omega_d=\sqrt{\omega_0^2-\gamma^2}
\]
Then:
\[
x(t)=e^{-\gamma t}\left(A\cos(\omega_d t)+B\sin(\omega_d t)\right)
\]
The constants \(A\) and \(B\) come from the initial displacement and velocity.
6. Series RLC circuit model
A free series RLC circuit has the equation:
\[
L q^{\prime\prime}+R q^\prime+\frac{1}{C}q=0
\]
Here:
- \(L\) is inductance,
- \(R\) is resistance,
- \(C\) is capacitance,
- \(q(t)\) is charge,
- \(i(t)=q^\prime(t)\) is current.
Dividing by \(L\):
\[
q^{\prime\prime}+\frac{R}{L}q^\prime+\frac{1}{LC}q=0
\]
7. RLC and spring-mass analogy
The RLC circuit has the same mathematical structure as a damped spring-mass system.
8. Population growth model
Logistic population growth is modeled by:
\[
P^\prime=rP\left(1-\frac{P}{K}\right)
\]
Here:
- \(P(t)\) is population,
- \(r\) is the growth rate,
- \(K\) is the carrying capacity.
The carrying capacity \(K\) is the stable long-term population level when \(r>0\).
9. Logistic solution
For initial population \(P(0)=P_0\), the logistic solution is:
\[
P(t)=\frac{K}{1+A e^{-rt}}
\]
where:
\[
A=\frac{K-P_0}{P_0}
\]
If \(00\), the population increases toward \(K\).
10. Mixing tank model
A standard mixing tank model tracks the amount of solute \(A(t)\) in a tank.
For a tank with constant volume \(V\), equal inflow and outflow rate \(q\), and incoming concentration
\(C_{\mathrm{in}}\), the model is:
\[
A^\prime=qC_{\mathrm{in}}-\frac{q}{V}A
\]
The term \(qC_{\mathrm{in}}\) is the solute entering per unit time.
The term \(\frac{q}{V}A\) is the solute leaving per unit time.
11. Mixing tank solution
The equilibrium amount is:
\[
A_{\mathrm{eq}}=VC_{\mathrm{in}}
\]
The solution is:
\[
A(t)=A_{\mathrm{eq}}+\left(A_0-A_{\mathrm{eq}}\right)e^{-\frac{q}{V}t}
\]
The tank concentration is:
\[
C(t)=\frac{A(t)}{V}
\]
12. Equilibrium and long-term behavior
Many physical models have an equilibrium level. The graph helps show whether the solution approaches,
crosses, or oscillates around this level.
14. Common mistakes
- Ignoring units: parameters must be consistent, such as seconds with seconds or minutes with minutes.
- Confusing damping with stiffness: \(c\) controls damping, while \(k\) controls spring strength.
- Confusing charge and current: in an RLC circuit, \(i(t)=q^\prime(t)\).
- Forgetting carrying capacity: logistic growth is not unlimited; it approaches \(K\).
- Using amount instead of concentration: in mixing, concentration is \(A(t)/V\).
- Forgetting initial conditions: constants in the solution depend on starting values.
- Reading the graph without units: the axis numbers only make sense with the selected units.
