Growth and Decay
Differential equations are pivotal in modelling natural phenomena, particularly in representing processes of growth and decay, which are ubiquitous in various scientific fields. For a deeper understanding of the mathematical principles underlying growth processes, reviewing exponential functions can provide valuable insights.
Exponential Growth and Decay
Exponential growth and decay are modelled by first-order linear differential equations. The general form for exponential growth and decay is given by:
dy/dt = ky
where:
- y(t) is the quantity at time t
- k is a constant; k > 0 represents growth, k < 0 represents decay
For those looking to expand their knowledge on the differentiation of functions relevant to growth and decay models, the differentiation of exponential and logarithmic functions offers an in-depth exploration.
Example 1: Population Growth
Consider a population P(t) that grows exponentially with a rate of 2% per year. The differential equation modelling this is:
dP/dt = 0.02P
To solve, we can separate variables and integrate, yielding the solution P(t) = P0e(0.02t), where P0 is the initial population.
Logistic Growth
Logistic growth is modelled by a first-order non-linear differential equation, representing limited growth:
dP/dt = rP(1 - P/K)
where:
- P(t) is the population at time t
- r is the relative growth rate
- K is the carrying capacity
Example 2: Limited Population Growth
If a population of bacteria has a carrying capacity of 1000 and a relative growth rate of 0.1, the logistic equation becomes:
dP/dt = 0.1P(1 - P/1000)
This equation can be solved using separation of variables and partial fraction decomposition.
Understanding the integration of trigonometric functions can also be beneficial when dealing with differential equations in biology and other fields.
Electrical Circuits
Differential equations are instrumental in analysing electrical circuits, particularly in describing the dynamics of current and voltage across resistors, capacitors, and inductors. Those interested in the basics of circuit analysis should consider the foundational concepts of basic differentiation rules.
RC Circuits
In an RC (resistor-capacitor) circuit, the voltage across the capacitor, V(t), is described by the first-order linear differential equation:
RC(dV/dt) + V = E
where:
- R is the resistance
- C is the capacitance
- E is the electromotive force (emf)
Example 3: Charging a Capacitor
Consider an RC circuit with R = 2 ohms, C = 1 F, and E = 10 V. The differential equation becomes:
2(dV/dt) + V = 10
This equation can be solved using an integrating factor, yielding the voltage across the capacitor as a function of time.
RLC Circuits
In an RLC (resistor-inductor-capacitor) circuit, the second-order linear differential equation describes the circuit:
L(d2Q/dt2) + R(dQ/dt) + Q/C = E(t)
where:
- L is the inductance
- Q(t) is the charge on the capacitor
For further exploration of circuits and their differential equations, the page on second-order differential equations offers comprehensive insights.
Example 4: Oscillations in an RLC Circuit
Consider an RLC circuit with L = 1 H, R = 1 ohm, C = 1 F, and E(t) = 10sin(t). The differential equation becomes:
(d2Q/dt2) + (dQ/dt) + Q = 10sin(t)
This non-homogeneous equation can be solved using methods like undetermined coefficients or variation of parameters.
Final Thoughts
Through the lens of differential equations, we gain insights into the underlying mechanisms of growth, decay, and electrical circuits, enabling us to mathematically model and analyse these dynamic systems. Engaging with these mathematical models not only enhances our understanding of the phenomena but also equips us with the analytical tools to predict future behaviour and devise strategies for control and optimization in various scientific and engineering contexts.
FAQ
Phase space and phase portraits provide a graphical representation of the solutions to differential equations, particularly useful for second-order linear differential equations like those in RLC circuits. In the context of electrical circuits, a phase portrait plots the charge Q and current I (dQ/dt) in a two-dimensional space for different initial conditions, illustrating the system's dynamic behaviour. Each point in this space represents a state of the circuit, and trajectories show how the charge and current evolve over time. Phase portraits help visualize stable and unstable equilibria and oscillatory behaviours of electrical circuits.
The Laplace Transform is a powerful method for solving linear differential equations, especially in electrical circuit analysis. It transforms the differential equation, which is a function of time, into an algebraic equation, which is a function of a complex variable s. This algebraic equation is often simpler to solve. Once solved, the Inverse Laplace Transform is used to convert the solution back into the time domain. Particularly in electrical circuits, the Laplace Transform method simplifies the analysis by allowing impedances to be used to represent resistors, capacitors, and inductors, facilitating the application of Ohm’s and Kirchhoff’s laws in the s-domain.
The method of separation of variables is a technique used to solve first-order differential equations like the logistic equation. The logistic equation is dP/dt = rP(1 - P/K). To apply separation of variables, we rearrange the equation to isolate all terms involving P on one side and t on the other: (1/P(1 - P/K))dP = rdt. Then, we integrate both sides separately. The left side requires partial fraction decomposition before integrating. This method provides a solution for P(t), describing how the population P changes with time t, given specific initial conditions and parameter values.
The characteristic equation arises when solving second-order linear homogeneous differential equations, like those found in RLC circuits, using the auxiliary m-method. The characteristic equation is derived by assuming a solution of the form Q(t) = e(mt) and substituting this into the differential equation, resulting in a quadratic equation in m. The roots of the characteristic equation, m1 and m2, determine the form of the general solution to the differential equation, influencing the behaviour of the electrical quantities (like charge, current, and voltage) in the circuit over time, and providing insights into the circuit’s transient response.
Equilibrium solutions in the logistic growth model refer to the population levels that remain constant over time, meaning dP/dt = 0. In the logistic differential equation, dP/dt = rP(1 - P/K), setting dP/dt = 0 and solving for P yields the equilibrium solutions P = 0 and P = K. These solutions are significant as they represent stable population levels. P = 0 is a trivial equilibrium, indicating no population. P = K is a stable equilibrium, representing the carrying capacity where the population remains steady, neither growing nor decaying, assuming no changes in the environment or parameters.
Practice Questions
The logistic growth model is given by the differential equation dP/dt = rP(1 - P/K), where P(t) is the population at time t, r is the relative growth rate, and K is the carrying capacity. Given P(0) = 100, P(3) = 200, and K = 1000, we can use these values to find r. First, we need to find the general solution of the logistic equation, which is P(t) = K / (1 + (K/P0 - 1)e(-rt)). Substituting P(3) = 200 into this equation, we can solve for r. Rearranging the equation, we get 200 = 1000 / (1 + (1000/100 - 1)e(-3r)). Solving for r, we find that r is approximately 0.37.
The differential equation for an RLC circuit with an alternating voltage source is L(d2Q/dt2) + R(dQ/dt) + Q/C = E(t), where L is the inductance, R is the resistance, C is the capacitance, and E(t) is the electromotive force. Substituting the given values, we get 2(d2Q/dt2) + (dQ/dt) + 2Q = 10sin(t). To solve this non-homogeneous differential equation, we can use the method of undetermined coefficients, assuming a particular solution of the form Qp(t) = A sin(t) + B cos(t), and substituting this into the differential equation to solve for A and B. The general solution is then Q(t) = Qh(t) + Qp(t), where Qh(t) is the general solution to the homogeneous equation 2(d2Q/dt2) + (dQ/dt) + 2Q = 0. This solution will provide the charge on the capacitor at any time t in terms of A, B, and any constants from the homogeneous solution.
