Differential Equations: Introduction and Formulation
Introduction to Differential Equations: Definition
In mathematics and its applications, many phenomena in physics, biology, chemistry, engineering, economics, etc., can be described by equations that involve not only the quantities themselves but also their rates of change. These equations are called differential equations. Differential equations are fundamental tools for modeling dynamic systems, i.e., systems that evolve or change over time or space.
What is a Differential Equation?
An equation is classified as a differential equation if it contains one or more derivatives of an unknown function. More formally, a differential equation is an equation that relates an independent variable, a dependent variable (which is a function of the independent variable), and one or more derivatives of the dependent variable with respect to the independent variable(s).
For example, if $y$ is a dependent variable and $x$ is an independent variable, a differential equation could involve $x$, $y$, $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, etc. If $u$ is a dependent variable and $x$ and $t$ are independent variables, a differential equation could involve $x$, $t$, $u$, $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial t}$, $\frac{\partial^2 u}{\partial x^2}$, etc.
In essence, a differential equation is an equation where the "unknown" is a function, and the equation includes the function's derivatives. Solving a differential equation means finding the function(s) that satisfy the given relationship between the function and its rates of change.
Types of Differential Equations
Differential equations are primarily classified based on the type of derivatives they contain, which in turn depends on the number of independent variables involved.
-
Ordinary Differential Equations (ODE):
- An equation is an Ordinary Differential Equation if it involves derivatives of a dependent variable with respect to only one independent variable.
- The derivatives are ordinary derivatives (like $\frac{dy}{dx}$, $\frac{d^2y}{dx^2}$, $\frac{d^3y}{dx^3}$, $y'$, $y''$, $y'''$, etc.). These are used when the dependent variable depends on a single independent variable.
- ODEs are used to model systems where the change depends on a single factor (often time or position).
Examples of ODEs:
- $\frac{dy}{dx} = 2x + 3$: Relates the first derivative of $y$ (w.r.t. $x$) to an expression in $x$.
- $\frac{d^2y}{dx^2} + 5 \frac{dy}{dx} + 6y = 0$: A second-order ODE relating a function and its first and second derivatives.
- $(y''')^2 + (y')^1 = \sin x$: Involves the first and third derivatives of $y$ w.r.t. $x$.
- $\frac{dr}{dt} = -kr$: Models radioactive decay or population growth (rate of change of $r$ w.r.t. $t$ is proportional to $r$). Here $r$ is dependent on $t$.
-
Partial Differential Equations (PDE):
- An equation is a Partial Differential Equation if it involves partial derivatives of a dependent variable with respect to two or more independent variables.
- Partial derivatives (like $\frac{\partial u}{\partial x}$, $\frac{\partial u}{\partial y}$, $\frac{\partial u}{\partial t}$, $\frac{\partial^2 u}{\partial x^2}$, $\frac{\partial^2 u}{\partial y^2}$, $\frac{\partial^2 u}{\partial t^2}$, etc.) are used when the dependent variable is a function of multiple independent variables (e.g., temperature $u$ depending on position $(x, y, z)$ and time $t$, $u(x, y, z, t)$).
- PDEs are used to model systems that vary in both space and time, or across multiple spatial dimensions (e.g., heat distribution, wave propagation, fluid dynamics).
Examples of PDEs:
- $\frac{\partial u}{\partial t} = k \frac{\partial^2 u}{\partial x^2}$: The one-dimensional Heat Equation, modeling how temperature $u$ changes over time $t$ and one spatial dimension $x$.
- $\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} = 0$: Laplace's Equation, describing steady-state phenomena like temperature distribution or electric potential in a plane. Here $z$ is dependent on $x$ and $y$.
- $\frac{\partial^2 y}{\partial t^2} = c^2 \frac{\partial^2 y}{\partial x^2}$: The one-dimensional Wave Equation, modeling the motion of waves (like a vibrating string). Here $y$ is dependent on position $x$ and time $t$.
In standard introductory calculus and differential equations courses, the primary focus is on Ordinary Differential Equations (ODEs) because they are simpler to solve and analyze before moving to PDEs.
Order and Degree of a Differential Equation
Differential equations are characterized by their order and, sometimes, their degree. These classifications help categorize equations and determine the appropriate methods for solving them.
Order of a Differential Equation
The order of a differential equation is a fundamental characteristic. It indicates the highest level of differentiation involved in the equation.
- Definition: The order of a differential equation is the order of the highest derivative that appears anywhere in the equation.
- How to Find the Order: Examine all the derivative terms in the equation (e.g., $\frac{dy}{dx}$ or $y'$, $\frac{d^2y}{dx^2}$ or $y''$, $\frac{d^3y}{dx^3}$ or $y'''$, $\frac{\partial u}{\partial x}$, $\frac{\partial^2 u}{\partial t^2}$). Identify the derivative with the largest order (1st, 2nd, 3rd, etc.). The order of this highest derivative is the order of the differential equation.
Examples:
- $\frac{dy}{dx} = 2x+3$: The highest derivative is $\frac{dy}{dx}$, which is a first derivative. The order is 1.
- $\frac{d^2y}{dx^2} + 5 \frac{dy}{dx} + 6y = 0$: The highest derivative is $\frac{d^2y}{dx^2}$, which is a second derivative. The order is 2.
- $y''' + (y')^4 = x^2$: The highest derivative is $y'''$, which is a third derivative. The order is 3. (The power of $y'$ does not affect the order).
- $\frac{\partial^2 u}{\partial x^2} + \frac{\partial u}{\partial t} = 0$: The highest derivative is $\frac{\partial^2 u}{\partial x^2}$, which is a second-order partial derivative. The order is 2.
Degree of a Differential Equation
The degree of a differential equation, when defined, refers to the power to which the highest-order derivative is raised. However, this definition only applies if the differential equation can be written as a polynomial in terms of the dependent variable and its derivatives.
- Definition: The degree of a differential equation is the highest power (exponent) of the highest order derivative appearing in the equation, provided that the equation is a polynomial equation in the dependent variable and its derivatives, and after it has been cleared of any radicals or fractional powers with respect to the derivatives.
- How to Find the Degree:
- Identify the highest order derivative present in the equation.
- Ensure that the equation can be expressed as a polynomial in the dependent variable ($y$) and its derivatives ($y', y'', y'''$, etc.). This means terms like $\sqrt{y'}$, $(y'')^{1/2}$, $\sin(y')$, $e^{y''}$, $\ln(y''')$ are generally not allowed if you want to determine the degree. If the equation contains radicals or fractional powers of derivatives, raise both sides of the equation to an appropriate power to eliminate them.
- Once the equation is a polynomial in derivatives (and cleared of radical/fractional powers of derivatives), find the highest power (exponent) to which the highest order derivative (identified in step 1) is raised. This power is the degree.
- Note: If the differential equation cannot be written as a polynomial in its derivatives (e.g., if it involves transcendental functions of derivatives like $\sin(y')$ or $e^{y''}$), the degree is generally considered to be not defined.
Examples Illustrating Order and Degree
| Differential Equation | Highest Order Derivative | Order | Equation as Polynomial in Derivatives? | Power of Highest Order Derivative | Degree |
|---|---|---|---|---|---|
| $\frac{dy}{dx} = \sin x$ | $\frac{dy}{dx}$ (1st order) | 1 | Yes (Implicitly $(\frac{dy}{dx})^1 = \sin x$) | 1 | 1 |
| $\frac{d^2y}{dx^2} + 5y = 0$ | $\frac{d^2y}{dx^2}$ (2nd order) | 2 | Yes (Implicitly $(\frac{d^2y}{dx^2})^1 + 5y^1 = 0$) | 1 | 1 |
| $(y''')^2 + (y'')^5 + y = e^x$ | $y'''$ (3rd order) | 3 | Yes | 2 (from $(y''')^2$) | 2 |
| $\left(\frac{d^2y}{dx^2}\right)^{3/2} = 1 + \frac{dy}{dx}$ | $\frac{d^2y}{dx^2}$ (2nd order) | 2 | No, due to power $3/2$. Clear the fractional power by squaring both sides: $\left(\frac{d^2y}{dx^2}\right)^3 = \left(1 + \frac{dy}{dx}\right)^2$. This is now a polynomial in derivatives. | 3 (from $(\frac{d^2y}{dx^2})^3$) | 3 |
| $\sqrt{1 + (y')^2} = y''$ | $y''$ (2nd order) | 2 | No, due to the square root. Clear the radical by squaring both sides: $1 + (y')^2 = (y'')^2$. This is now a polynomial in derivatives. | 2 (from $(y'')^2$) | 2 |
| $\sin\left(\frac{dy}{dx}\right) + x = 0$ | $\frac{dy}{dx}$ (1st order) | 1 | No. The term $\sin(\frac{dy}{dx})$ means the equation is not a polynomial in terms of the derivative $\frac{dy}{dx}$. | N/A | Not Defined |
Determining the order is always possible for any differential equation involving derivatives. Determining the degree is only possible if the equation is a polynomial in its derivatives.
Solution of Differential Equation (General, Particular, Singular)
Solving a differential equation means finding the function(s) that satisfy the equation. Unlike algebraic equations whose solutions are typically numbers, the solutions of differential equations are functions or relations between variables. Depending on whether arbitrary constants are involved, the solutions are classified as general, particular, or singular.
Definition of a Solution
A solution (or integral or primitive) of a differential equation is a function $y = \phi(x)$ (or a relation $G(x, y) = 0$) that, when substituted into the differential equation along with its corresponding derivatives, satisfies the equation identically for all values of the independent variable $x$ in some interval.
In simpler terms, a solution is a function that "works" when plugged into the differential equation. It should contain no derivatives and should make the equation true for all relevant values of the independent variable.
Types of Solutions
Differential equations can have different types of solutions:
-
General Solution:
- The general solution of a differential equation of order $n$ is a solution that contains exactly $n$ arbitrary independent constants. These constants can take on any real value.
- The general solution represents a family of curves. Each specific curve in this family corresponds to a unique combination of values for the arbitrary constants.
- Finding the general solution often involves integrating the differential equation $n$ times, introducing one constant of integration at each integration step.
- Example: For the first-order ODE $\frac{dy}{dx} = 2x$, we integrate both sides with respect to $x$: $\int \frac{dy}{dx} dx = \int 2x dx \implies y = x^2 + C$. This is the general solution, containing one arbitrary constant $C$, which equals the order of the ODE (1).
- Example: For the second-order ODE $\frac{d^2y}{dx^2} = 6x$, integrating once gives $\frac{dy}{dx} = \int 6x dx = 3x^2 + C_1$. Integrating a second time gives $y = \int (3x^2 + C_1) dx = x^3 + C_1 x + C_2$. This is the general solution, containing two arbitrary constants $C_1$ and $C_2$, which equals the order of the ODE (2).
-
Particular Solution:
- A particular solution is a solution obtained from the general solution by assigning specific, fixed numerical values to the arbitrary constants.
- These specific values are determined by using additional information, usually provided as initial conditions or boundary conditions.
- An initial condition specifies the value of the dependent variable and/or its derivatives at a single point of the independent variable (often at a starting time, $t=0$). For an $n$-th order ODE, you typically need $n$ independent initial conditions to uniquely determine the $n$ arbitrary constants in the general solution. Example: For $y'' = f(x, y, y')$, conditions could be $y(x_0)=y_0, y'(x_0)=y_1$.
- A boundary condition specifies the value of the dependent variable and/or its derivatives at two or more different points of the independent variable (often at the boundaries of an interval). Boundary conditions are common in problems defined over spatial intervals.
- Example: For the general solution $y = x^2 + C$ of $\frac{dy}{dx}=2x$, if we are given the initial condition $y(1)=3$, we substitute $x=1$ and $y=3$ into the general solution: $3 = (1)^2 + C \implies 3 = 1 + C \implies C = 2$. The particular solution satisfying the condition is $y = x^2 + 2$.
-
Singular Solution:
- A singular solution is a solution that satisfies the differential equation but cannot be obtained from the general solution by assigning any specific, finite numerical values to the arbitrary constants in the general solution.
- Singular solutions often arise in non-linear differential equations. Geometrically, they may represent the envelope of the family of curves described by the general solution.
- Example: Consider the differential equation $(y')^2 - 4y = 0$. The general solution is $y = (x+C)^2$. To check this, $y' = 2(x+C)$, and $(y')^2 - 4y = (2(x+C))^2 - 4(x+C)^2 = 4(x+C)^2 - 4(x+C)^2 = 0$. However, the function $y=0$ also satisfies the original differential equation: $y'=0$, so $(0)^2 - 4(0) = 0$. But $y=0$ cannot be obtained from $y=(x+C)^2$ for any finite value of $C$. Therefore, $y=0$ is a singular solution.
In most introductory settings focused on linear ODEs, singular solutions are not encountered, and the focus is on finding general and particular solutions.
Formation of Differential Equations (from Family of Curves)
The process of solving a differential equation involves finding the function(s) (or family of functions) that satisfy it. The reverse process is also possible: given a description of a family of curves (typically an equation involving variables and arbitrary constants), we can find the differential equation for which this family is the general solution.
This process involves eliminating the arbitrary constants from the equation describing the family of curves by using differentiation.
Process of Forming a Differential Equation from a Family of Curves
If an equation representing a family of curves involves $n$ arbitrary independent constants, the corresponding differential equation is obtained by eliminating these $n$ constants. This generally requires differentiating the original equation $n$ times.
Steps:
- Identify Arbitrary Constants: Examine the given equation of the family of curves and identify the arbitrary constants. Count the number of such independent arbitrary constants, let this number be $n$. (Constants like $\pi$ or $e$ are not arbitrary constants in this context; they are specific numbers).
- Differentiate $n$ Times: Differentiate the given equation $n$ times successively with respect to the independent variable (usually $x$, unless otherwise specified). This will generate $n$ new equations involving the variables, the original constants, and derivatives up to order $n$. You will now have a total of $n+1$ equations (the original equation plus the $n$ equations obtained by differentiation).
- Eliminate Arbitrary Constants: Use algebraic manipulation to eliminate the $n$ arbitrary constants from the system of $n+1$ equations obtained in Step 2. This often involves substituting expressions for the constants or derivatives from some equations into others.
- Formulate the Differential Equation: The equation that results from the elimination process, which contains only the variables (dependent and independent) and their derivatives (and is free of the arbitrary constants), is the required differential equation. The order of this resulting differential equation will generally be equal to the number of arbitrary constants $n$ you started with.
Example 1. Form the differential equation representing the family of straight lines $y = mx$, where $m$ is an arbitrary constant.
Answer:
Step 1: Identify Arbitrary Constants.
The given equation is $y = mx$. The term $m$ is an arbitrary constant. There is only one such constant. So, $n=1$. We expect a first-order differential equation.
"$y = mx$"
... (i)
Step 2: Differentiate $n=1$ time with respect to $x$.
Differentiate equation (i) with respect to $x$. Remember that $m$ is a constant with respect to $x$.
"$\frac{dy}{dx} = \frac{d}{dx}(mx) = m \frac{d}{dx}(x) = m(1) = m$"
[Differentiate (i)]
"$\frac{dy}{dx} = m$"
... (ii)
Step 3: Eliminate Arbitrary Constants.
We have two equations: (i) $y = mx$ and (ii) $\frac{dy}{dx} = m$. We need to eliminate the single arbitrary constant $m$ using these two equations.
From equation (ii), we have an expression for $m$ in terms of the derivative. Substitute this expression for $m$ into equation (i):
"$y = \left(\frac{dy}{dx}\right) x$"
[Substitute (ii) into (i)]
Step 4: Formulate the Differential Equation.
The resulting equation $y = x \frac{dy}{dx}$ contains only the variables $x$ and $y$ and the derivative $\frac{dy}{dx}$. It is free of the arbitrary constant $m$.
Rearranging into a standard form:
"$x \frac{dy}{dx} - y = 0$"
This is the required differential equation. Its highest derivative is $\frac{dy}{dx}$ (order 1), which matches the number of arbitrary constants ($n=1$).
The differential equation for the family of lines $y=mx$ is $x \frac{dy}{dx} - y = 0$.
Example 2. Find the differential equation of the family of circles $x^2 + y^2 = a^2$, where $a$ is an arbitrary constant.
Answer:
Step 1: Identify Arbitrary Constants.
The given equation is $x^2 + y^2 = a^2$. The term $a$ is an arbitrary constant (representing the radius). There is one arbitrary constant $a$. So, $n=1$. We expect a first-order differential equation.
"$x^2 + y^2 = a^2$"
... (i)
Step 2: Differentiate $n=1$ time with respect to $x$.
Differentiate equation (i) with respect to $x$. Remember that $y$ is a function of $x$ (implicitly), so we use implicit differentiation for terms involving $y$. $a^2$ is a constant, so its derivative is 0.
"$\frac{d}{dx}(x^2) + \frac{d}{dx}(y^2) = \frac{d}{dx}(a^2)$"
[Differentiate (i) w.r.t $x$]
Using the Power Rule and Chain Rule for $y^2$:
"$2x + 2y \frac{dy}{dx} = 0$"
[Applying rules]
Divide by 2 to simplify:
"$x + y \frac{dy}{dx} = 0$"
... (ii)
Step 3: Eliminate Arbitrary Constants.
We have two equations: (i) $x^2 + y^2 = a^2$ and (ii) $x + y \frac{dy}{dx} = 0$. We need to eliminate the constant $a$. However, equation (ii) already does not contain the arbitrary constant $a$. The process of differentiating once was sufficient to eliminate it.
Step 4: Formulate the Differential Equation.
Equation (ii) is the required differential equation:
"$x + y \frac{dy}{dx} = 0$"
This is a first-order ODE, matching the number of arbitrary constants ($n=1$).
The differential equation for the family of circles centered at the origin is $x + y \frac{dy}{dx} = 0$.
Differential Equations (Applied Maths Definition)
From an applied mathematics perspective, differential equations are more than abstract mathematical constructs; they are essential tools for building mathematical models that describe dynamic systems in the real world. They capture the laws and principles that govern how quantities change.
Key Aspects of Differential Equations in Applied Contexts
In applied fields, differential equations serve as the language to formulate and analyze problems involving rates of change and accumulation:
- Modeling Change: The fundamental idea is to translate verbal descriptions of how quantities change into mathematical equations using derivatives. For example, "the rate of growth of a population is proportional to its current size" translates to a differential equation $\frac{dP}{dt} = kP$.
- Describing Dynamic Systems: DEs model systems that evolve over time (like motion, chemical reactions, population dynamics) or vary across space (like heat distribution, stress in materials, fluid flow). The derivatives in the equation represent the dynamics of the system.
- Encoding Laws and Principles: Differential equations often directly embody fundamental laws of nature or established principles in various domains. Examples include Newton's laws of motion (relating force to acceleration, a second derivative), Fourier's law of heat conduction (relating heat flow to temperature gradients, derivatives), or economic growth models.
- Predictive and Analytical Tool: Solving a differential equation, along with given initial or boundary conditions (which describe the state of the system at a particular point in time or space), allows applied mathematicians to find the function that describes the system's behavior. This function can then be used to predict future states, analyze stability, or understand how system parameters influence the outcome.
- Foundation for Simulation: When analytical solutions are difficult or impossible, differential equations are the basis for numerical methods and computer simulations used to approximate the behavior of complex systems.
In essence, applied mathematics views differential equations as the primary mathematical tool for formulating and solving problems where the relationships between quantities are expressed in terms of their rates of change. The focus is on building models, analyzing their properties, and using them to gain insights into real-world phenomena.