Menu Top
Complete Course of Mathematics
Topic 1: Numbers & Numerical Applications Topic 2: Algebra Topic 3: Quantitative Aptitude
Topic 4: Geometry Topic 5: Construction Topic 6: Coordinate Geometry
Topic 7: Mensuration Topic 8: Trigonometry Topic 9: Sets, Relations & Functions
Topic 10: Calculus Topic 11: Mathematical Reasoning Topic 12: Vectors & Three-Dimensional Geometry
Topic 13: Linear Programming Topic 14: Index Numbers & Time-Based Data Topic 15: Financial Mathematics
Topic 16: Statistics & Probability


Content On This Page
Modeling with Differential Equations Application of Differential Equations in Various Fields (Growth, Decay, etc.) Formulating and Solving Differential Equations (Applied Maths)
Application of Differential Equations (Applied Maths)

Differential Equations: Modeling and Applications

Modeling with Differential Equations

One of the most powerful applications of calculus is its use in mathematical modeling. Mathematical modeling is the process of representing a real-world phenomenon using mathematical concepts, language, and tools. When the phenomenon involves quantities that change and the relationships governing these changes (i.e., rates of change), differential equations often provide the most natural and effective way to build a mathematical model.

A differential equation serves as a mathematical statement that describes the fundamental principles or laws governing the behavior of a system by relating the quantities in the system to their rates of change. Solving the differential equation allows us to predict or understand the behavior of the system over time or space.

The Concept of Modeling with Differential Equations

The core idea is to translate a real-world problem description into a mathematical equation involving derivatives. This requires identifying the key quantities that vary and formulating a relationship between them based on a principle or law. For example:

These verbal statements about rates of change can be written as differential equations. Once the differential equation is formulated, the techniques of solving differential equations (analytical or numerical) can be used to find the function that describes the quantity's behavior over time or space.

The Mathematical Modeling Process using Differential Equations

Creating and using a differential equation model to solve a real-world problem typically involves the following steps:

  1. Identify Variables and State Assumptions:
    • Determine the independent variable (often time, $t$, but could be position, $x$, or others).
    • Identify the dependent variable(s) – the quantity or quantities whose behavior you are interested in (e.g., population size $P$, temperature $T$, concentration $C$, amount of substance $A$, position $s$).
    • State any simplifying assumptions you make about the real-world system. Models are often simplifications of reality. Assumptions might include things like assuming a continuous process, neglecting factors like friction or air resistance, assuming constant rates, etc.
  2. Formulate the Governing Law or Principle:
    • Express the fundamental rule or principle that governs the change in the dependent variable. This rule should describe the rate of change of the dependent variable.
    • This might be based on established scientific laws (Newton's Laws, Conservation Laws), empirical observations, or logical reasoning about how the system changes. Often, this involves relating the rate of change to the current state of the system or other factors. A common framework is "Rate of Change = Rate In - Rate Out".
  3. Translate the Principle into a Differential Equation:
    • Write the relationship formulated in Step 2 using mathematical notation. The rate of change of the dependent variable is represented by its derivative(s) with respect to the independent variable.
    • This results in a differential equation involving the independent variable, the dependent variable, and its derivative(s).
  4. Specify Initial and/or Boundary Conditions:
    • A differential equation typically has a general solution representing a family of possible behaviors. To pinpoint the specific behavior of the system in a particular scenario, you need additional information about the state of the system at certain points.
    • Initial Conditions: These specify the value(s) of the dependent variable and/or its derivatives at a single value of the independent variable (e.g., $y(0) = y_0$, $y'(0) = v_0$, representing the state at the beginning of a process).
    • Boundary Conditions: These specify the value(s) of the dependent variable or its derivatives at different values of the independent variable, usually at the boundaries of an interval (e.g., $y(a) = y_a$, $y(b) = y_b$, for a problem defined over a spatial interval $[a, b]$).
    • These conditions are essential for finding the particular solution of the differential equation.
  5. Solve the Differential Equation:
    • Use appropriate mathematical techniques to find the general solution of the differential equation. This might involve methods like separation of variables, using integrating factors, etc., depending on the type and order of the DE.
  6. Apply Initial/Boundary Conditions:
    • Substitute the given initial or boundary conditions into the general solution (and its derivatives, if necessary) to determine the values of the arbitrary constants introduced during the integration process. This yields the particular solution that describes the specific behavior of the system under the given conditions.
  7. Validate and Interpret the Solution:
    • Check if the particular solution makes physical or practical sense in the context of the real-world problem. Does it exhibit the expected qualitative behavior?
    • Compare the model's predictions with observed data if available (validation).
    • Interpret the solution to answer the questions posed in the original problem. Analyze the influence of parameters in the model. Understand the limitations of the model based on the initial assumptions.

Differential equations provide the bridge between a qualitative understanding of how a quantity changes and a quantitative prediction of its future or overall behavior. They are thus indispensable in scientific and engineering analysis.


Application of Differential Equations in Various Fields (Growth, Decay, etc.)

Differential equations are fundamental tools for modeling dynamic systems across a wide spectrum of scientific, engineering, economic, and biological disciplines. They allow us to move from understanding the rates of change of quantities to predicting their actual values over time or space.

Common Applications and Examples

Here are some illustrative examples of how differential equations are applied in various fields:

  1. Population Dynamics (Biology, Ecology): Differential equations are used to model how the number of individuals in a population changes over time.

    • Exponential Growth/Decay: The simplest model assumes the rate of population change is directly proportional to the current population size $P(t)$.

      $\frac{dP}{dt} = kP$

      Here, $k$ is the proportionality constant (growth rate). If $k>0$, the population grows exponentially; if $k<0$, it decays exponentially (e.g., radioactive decay of a substance). The solution to this first-order separable ODE is $P(t) = P_0 e^{kt}$, where $P_0 = P(0)$ is the initial population size.

    • Logistic Growth: A more realistic model for limited environments, including a carrying capacity $L$ (the maximum sustainable population). The growth rate slows down as the population approaches $L$.

      $\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)$

      This is a first-order nonlinear ODE. Its solution exhibits S-shaped (logistic) growth.

    • Predator-Prey Models (Lotka-Volterra equations): A system of two coupled first-order nonlinear ODEs modeling the dynamic interaction between the populations of a predator species and a prey species.

  2. Physics: Many fundamental laws of physics are expressed as differential equations.

    • Newton's Law of Cooling: Describes how the temperature $T(t)$ of an object changes over time $t$ due to heat exchange with its surroundings at temperature $T_{env}$.

      $\frac{dT}{dt} = k(T - T_{env})$

      Here, $k$ is a cooling constant ($k<0$). This is a first-order linear ODE.

    • Newton's Second Law of Motion: Relates the force $F$ acting on an object of mass $m$ to its acceleration $a(t)$, which is the second derivative of its position $x(t)$.

      $F = ma = m \frac{d^2x}{dt^2}$

      This is a second-order ODE. Adding forces like gravity, friction, air resistance, or spring forces leads to various second-order linear or nonlinear ODEs describing motion.

    • Electrical Circuits (RLC Circuits): Kirchhoff's laws applied to circuits with resistors (R), inductors (L), and capacitors (C) often result in second-order linear ODEs describing the charge or current as a function of time.
    • Simple Harmonic Motion: Describes ideal oscillations (like a mass on a spring or a simple pendulum for small angles).

      $\frac{d^2x}{dt^2} = -\omega^2 x$

      This is a second-order linear homogeneous ODE, with solutions involving sines and cosines.

  3. Chemistry: Differential equations are used extensively in chemical kinetics and modeling concentrations.

    • Reaction Rates: The Law of Mass Action states that the rate of a chemical reaction is proportional to the product of the concentrations of the reactants, each raised to a power equal to its stoichiometric coefficient. This translates directly into differential equations describing how reactant and product concentrations change over time. Example: For $A \to P$, if rate is $k[A]^1$, $\frac{d[A]}{dt} = -k[A]$.
    • Mixing Problems: Modeling the amount of a substance (solute) in a tank where solutions flow in and out at specific rates, often involving a first-order linear ODE.

  4. Engineering: DEs are fundamental to virtually all branches of engineering for analyzing and designing systems.

    • Heat Transfer: The heat equation ($\frac{\partial T}{\partial t} = \alpha \nabla^2 T$) is a classic PDE modeling temperature distribution in space and time.
    • Fluid Dynamics: The Navier-Stokes equations are a set of coupled nonlinear PDEs that describe the motion of viscous fluids.
    • Structural Analysis: Modeling the deflection of beams, the vibration of structures, or the behavior of control systems often involves ODEs or PDEs.

  5. Economics and Finance: DEs are used to model changes in economic variables over time.

    • Compound Interest: Continuous compounding leads to the DE $\frac{dA}{dt} = rA$, where $A(t)$ is the amount of capital, and $r$ is the interest rate. Solution is $A(t) = A_0 e^{rt}$.
    • Economic Growth Models: Macroeconomic models relate the rates of change of national income, capital, labor, etc.
    • Option Pricing: The Black-Scholes equation is a famous second-order linear PDE used in financial mathematics to price options.

These examples highlight the widespread applicability of differential equations as a universal language for describing dynamic change and enabling predictions in systems where rates of change are central to the underlying processes.


Formulating and Solving Differential Equations (Applied Maths)

From an applied mathematics perspective, the power of differential equations lies in their ability to serve as mathematical models for real-world phenomena. The focus in applied settings is not just on solving the equation by applying a known technique, but also on the crucial initial step of formulating the correct differential equation that captures the dynamics of the system being studied, and finally, interpreting the mathematical solution back in the context of the original problem.

Emphasis in Formulation

Formulating a differential equation from a real-world description requires careful translation of concepts about rates of change into mathematical language. Key aspects include:

Emphasis in Solving

Once the differential equation is formulated, the focus shifts to finding its solution. From an applied perspective, this involves selecting and applying the appropriate technique accurately.

Example 1. A radioactive substance decays at a rate proportional to the amount present. If 100 mg of the substance are present initially, and after 10 years, 50 mg remain, find an expression for the amount remaining at any time $t$.

Answer:

This problem involves modeling a process (radioactive decay) using a differential equation and then solving it with given conditions.

Step 1: Identify Variables.

Independent variable: Time, $t$ (measured in years).

Dependent variable: Amount of radioactive substance, $A(t)$ (measured in milligrams, mg).

Step 2: Formulate the Rate Relationship.

The problem states that the substance decays at a rate proportional to the amount present. The rate of change of the amount $A$ with respect to time $t$ is $\frac{dA}{dt}$. "Proportional to the amount present" means $\frac{dA}{dt} \propto A$. The constant of proportionality is typically denoted by $k$. Since it is decay, the amount is decreasing, so the rate of change is negative. Thus, the constant of proportionality must be negative. Let's write the relationship as $\frac{dA}{dt} = kA$, where $k$ is a constant. Since decay occurs, $k$ will be negative. We can also write it as $\frac{dA}{dt} = -\lambda A$, where $\lambda$ is a positive constant related to the decay rate.

Using $k$ as the constant of proportionality:

"$\frac{dA}{dt} = kA$"

[Rate is proportional to amount]

Where $k < 0$ for decay.

Step 3: Write the Differential Equation.

The differential equation modeling the situation is already formulated from Step 2:

"$\frac{dA}{dt} = kA$"

This is a first-order ordinary differential equation.

Step 4: Identify Initial/Boundary Conditions.

The problem provides information about the amount of substance at specific times:

  • "100 mg of the substance are present initially": At $t=0$, $A = 100$ mg. This is an initial condition: $A(0) = 100$.
  • "after 10 years, 50 mg remain": At $t=10$ years, $A = 50$ mg. This is another condition: $A(10) = 50$.

We have one first-order ODE and two conditions. The first condition $A(0)=100$ is an initial condition that will allow us to determine the constant of integration. The second condition $A(10)=50$ will allow us to determine the decay constant $k$.

Step 5: Solve the Differential Equation.

The equation $\frac{dA}{dt} = kA$ is a variable separable differential equation. Assume $A \neq 0$.

Separate variables $A$ and $t$. Divide by $A$ and multiply by $dt$:

"$\frac{dA}{A} = k dt$"

[Separate variables]

Integrate both sides:

"$\int \frac{dA}{A} = \int k dt$"

Evaluate the integrals. $\int \frac{dA}{A} = \ln |A|$ and $\int k dt = kt$. Since $A$ represents the amount of substance, it must be positive, so $|A| = A$.

"$\ln A = kt + C_1$"

[Perform integration, add $C_1$]

Solve for $A$ by exponentiating both sides using base $e$:

"$A = e^{kt + C_1} = e^{kt} e^{C_1}$"

Let $A_0 = e^{C_1}$. Since $C_1$ is an arbitrary constant, $A_0$ is an arbitrary positive constant ($e^{C_1}$ is always positive). This gives the general solution:

"$A(t) = A_0 e^{kt}$"

[General solution]

Step 6: Apply Initial/Boundary Conditions to find Particular Solution.

Use the initial condition $A(0) = 100$ to find $A_0$. Substitute $t=0$ and $A=100$ into the general solution:

"$100 = A_0 e^{k(0)} = A_0 e^0 = A_0(1) = A_0$"

So, $A_0 = 100$. This means the constant $A_0$ represents the initial amount of the substance. The solution now is $A(t) = 100 e^{kt}$.

Now use the condition $A(10) = 50$ to find the decay constant $k$. Substitute $t=10$ and $A=50$ into $A(t) = 100 e^{kt}$:

"$50 = 100 e^{k(10)}$"

Divide by 100:

"$\frac{50}{100} = e^{10k} \implies \frac{1}{2} = e^{10k}$"

Take the natural logarithm of both sides to solve for $10k$:

"$\ln\left(\frac{1}{2}\right) = \ln(e^{10k})$"

Using logarithm properties ($\ln(1/2) = \ln 1 - \ln 2 = 0 - \ln 2 = -\ln 2$ and $\ln(e^{10k}) = 10k$):

"$-\ln 2 = 10k$"

Solve for $k$:

"$k = -\frac{\ln 2}{10}$"

As expected for decay, $k$ is negative. Now substitute the value of $k$ back into the solution $A(t) = 100 e^{kt}$.

Step 7: Final Solution (Expression for Amount Remaining).

Substitute $k = -\frac{\ln 2}{10}$ into $A(t) = 100 e^{kt}$:

"$A(t) = 100 e^{\left(-\frac{\ln 2}{10}\right) t}$"

This is a valid expression. We can simplify it using exponent and logarithm properties: $e^{ab} = (e^a)^b$ and $e^{\ln u} = u$.

"$A(t) = 100 e^{\ln 2 \cdot \left(-\frac{t}{10}\right)} = 100 \left(e^{\ln 2}\right)^{-\frac{t}{10}} = 100 (2)^{-\frac{t}{10}}$"

The expression for the amount of radioactive substance remaining at any time $t$ is $A(t) = 100 (2)^{-t/10}$ mg or equivalently $A(t) = 100 e^{-(\frac{\ln 2}{10})t}$ mg.

Interpretation: The half-life of the substance is 10 years (amount is halved every 10 years). The initial amount is 100 mg. The base 2 in the first form directly shows the halving based on the number of 10-year periods ($t/10$).


Application of Differential Equations (Applied Maths)

This perspective emphasizes the role of differential equations as essential tools for understanding and predicting the behavior of systems in various scientific and engineering domains.

In applied mathematics and related fields, the focus shifts from simply solving a differential equation to using it as a model to analyze a real-world system. The solution itself is a step towards understanding the system's dynamics, predicting its future state, or controlling its behavior.

Key Focus Areas in Applied Differential Equations

Beyond the steps of formulation and solution, the applied use of differential equations involves a deeper analysis of the solution and the model:

Applied mathematics uses differential equations not just as equations to be solved, but as dynamic descriptions of reality. They enable us to simulate, predict, design, and control complex systems across science, engineering, economics, business, and other fields by capturing the underlying laws of change.