First-order Differential Equations

Definition of First-order Differential Equations

In this general form, \(F\) is a function of three variables: \(x\) (the independent variable), \(y\) (the dependent variable), and \(\frac{dy}{dx}\) (the first derivative of \(y\) with respect to \(x\)). The primary goal when working with a first-order differential equation is to find the function \(y(x)\), which satisfies the given equation.

Types of First-order Differential Equations

There are various types of first-order differential equations, each with different techniques for solving them. In this section, we'll explore two of the most common types: homogeneous and separable first-order differential equations.

First-order Homogeneous Differential Equation

To solve a homogeneous first-order differential equation, you can use the substitution method. This involves introducing a new variable \(v = \frac{y}{x}\), which leads to a separable differential equation that is easier to solve. Once the new variable's equation has been determined, you can revert back to the original variables to find the solution for \(y(x)\).

Separable First-order Differential Equations

To solve a separable first-order differential equation, you can follow these steps: 1. Separate the variables by dividing both sides of the equation by \(H(y)\), and multiplying both sides by \(dx\), obtaining the following equation: \[\frac{dy}{H(y)} = G(x) \,dx\] 2. Integrate both sides of the equation with respect to their respective variables: \[\int \frac{dy}{H(y)} = \int G(x) \,dx\] 3. Solve the integrals and combine the resulting expressions to find the solution to the original differential equation.In conclusion, first-order differential equations are essential concepts in further mathematics. Understanding the definition, properties, and various types of first-order differential equations will help you excel in your studies and increase your problem-solving skills in various scientific and engineering disciplines.

Solving First-order Differential Equations

To solve first-order differential equations, various methods can be employed depending on the type and complexity of the equation. As mentioned earlier, techniques such as substitution and separation of variables can help you find the function \(y(x)\), which satisfies the given equation. In this section, we will expound on the general solution of a first-order differential equation and showcase some examples of different types of first-order equations.

General Solution of First-order Differential Equation

The general solution of a first-order differential equation represents a family of possible solutions. It typically contains a constant, denoted as 'c', which can have different values, leading to different specific solutions in the family. In some cases, a first-order differential equation may have more than one distinct general solution based on its form. To find the general solution of a first-order differential equation, you can take the following steps: 1. Identify the type of the equation, such as homogeneous or separable, to determine which method to apply. 2. Apply the appropriate method for solving the identified type of differential equation, such as substitution for homogeneous or separation of variables for separable equations. 3. Solve the resulting simplified equation, such as integrating both sides of a separable differential equation. 4. Find the general solution by determining an expression for the dependent variable, \(y(x)\), in terms of the independent variable, \(x\), and the constant 'c. Keep in mind that while finding the general solution provides an overview of possible solutions, you may need a specific solution in certain cases. To obtain a specific solution, you would use an initial condition, which will provide a point on the curve on which your solution lies. Using this initial condition, you can determine the particular value for the constant 'c' in the general solution.

First-order Differential Equation Example

Let's look at an example of solving a first-order differential equation: Consider the first-order differential equation: \[\frac{dy}{dx} + 2y = x\] This equation is neither homogeneous nor separable, but you can still solve it using an integrating factor, a technique commonly used for linear first-order differential equations. The general form for such linear equations is: \[\frac{dy}{dx} + p(x)y = q(x)\] Here, the integrating factor is given by the expression: \[I(x) = e^{\int p(x) \, dx}\] Follow these steps to solve the equation: 1. Identify the values of \(p(x)\) and \(q(x)\) in the given equation. Here, \(p(x) = 2\) and \(q(x) = x\). 2. Calculate the integrating factor \(I(x)\): \[I(x) = e^{\int 2\, dx} = e^{2x}\] 3. Multiply both sides of the differential equation by the integrating factor: \[\frac{dy}{dx} e^{2x} + 2ye^{2x} = xe^{2x}\] 4. The left side of the equation is now the product rule of differentiation of the product of \(y\) and \(I(x)\): \[\frac{d}{dx}(ye^{2x}) = xe^{2x}\] 5. Integrate both sides of the equation with respect to \(x\): \[\int d(ye^{2x}) = \int xe^{2x} \, dx\] 6. Solve the integrals: \[ye^{2x} = \frac{1}{4} xe^{2x} - \frac{1}{8} e^{2x} + C\] 7. Isolate the dependent variable \(y\) to obtain the general solution: \[y(x) = \frac{1}{4}x - \frac{1}{8} + Ce^{-2x}\] Note that the general solution contains a constant 'C', which can take any value to represent the different solutions within the family of solutions.

First-order Homogeneous Differential Equation Examples

Let's explore some examples of first-order homogeneous differential equations to solidify your understanding. 1. Solve the homogeneous first-order differential equation: \[\frac{dy}{dx} = \frac{2x - y}{x + y}\] To approach this problem, perform the following steps: - Use the substitution method by setting \(v = \frac{y}{x}\) and subsequently determining \(y = xv\) - Replace \(\frac{dy}{dx}\) in the equation with the equivalent expression in terms of \(x\) and \(v\) - Separate the variables and integrate each side of the equation - Revert to the original variables, \(x\) and \(y\), to find the general solution 2. Solve the homogeneous first-order differential equation: \[\frac{dy}{dx} = \frac{y^2}{x^2}\] To solve this equation: - Apply the substitution \(v = \frac{y}{x}\) to simplify the equation, similar to the previous example - Separate the variables and integrate both sides of the equation - Revert to the original variables to obtain the final general solution Through practice and better understanding of the various methods for solving first-order differential equations, you'll be well-equipped to handle complex problems throughout your studies in mathematics and its applications in numerous fields.

Laplace Transform of First-order Differential Equation

The Laplace Transform is a powerful tool that can simplify the process of solving linear differential equations. It converts a differential equation in the time domain into an algebraic equation in a transformed (s-domain) that is generally easier to solve. Once the equation is solved in the s-domain, the Inverse Laplace Transform is employed to revert the solution to the time domain. Applying the Laplace Transform to a first-order differential equation involves the following steps: 1. Write down the first-order differential equation. 2. Apply the Laplace Transform to both sides of the equation. 3. Rearrange the transformed equation into an algebraic form that is simpler to solve. 4. Solve the algebraic equation for the transformed dependent variable. 5. Use the Inverse Laplace Transform to obtain the solution of the original differential equation in the time domain. Consider the following first-order differential equation with an initial condition: \[\frac{dy}{dt} + ay = f(t), \quad y(0) = y_0\] Applying the Laplace Transform, we get: \[sY(s) - y_0 + aY(s) = F(s)\] where \(Y(s)\) and \(F(s)\) are the Laplace Transforms of \(y(t)\) and \(f(t)\), respectively. Solve for \(Y(s)\) to get: \[Y(s) = \frac{y_0 + F(s)}{s + a}\] Finally, apply the Inverse Laplace Transform to find the solution \(y(t)\) in the time domain.

Application of First-order Differential Equations

First-order differential equations have widespread applications in various disciplines. They are used to model systems that change over time, making them essential in understanding real-world phenomena. The following examples demonstrate some of the key applications of first-order differential equations: 1. Physics: In physics, first-order differential equations are employed to model a variety of processes, including Newton's Law of Cooling, radioactive decay, and charging or discharging of capacitors in electrical circuits. For example, the rate of change of temperature in an object subjected to a constant ambient temperature can be described by the following first-order equation: \[\frac{dT}{dt} = -k(T - T_{ambient})\] where \(T\) is the object's temperature, \(T_{ambient}\) is the constant ambient temperature, and \(k\) is a positive constant representing the cooling rate. 2. Chemistry: First-order differential equations are crucial for understanding chemical reactions. They can determine reaction rates and the concentration of substances over time. The common application is in modelling first-order reactions, which follow the equation: \[-\frac{d[A]}{dt} = k[A]\] Here, \([A]\) denotes the concentration of the reactant A, and \(k\) is the rate constant. 3. Population Dynamics: In biology, first-order differential equations can help study population growth, decay, or predator-prey interactions. The exponential growth or decay of a population can be represented by the following first-order equation: \[\frac{dP}{dt} = kP\] where \(P\) is the population at time \(t\), and \(k\) is the growth rate constant. 4. Economics: First-order differential equations play a significant role in economics, especially in determining growth rates and modelling investment decisions. A well-known application is the Solow growth model in macroeconomics, which can be presented in the form of a first-order differential equation: \[\frac{dK}{dt} = sY - (n + g + \delta)K\] Here, \(K\) denotes capital stock, \(Y\) represents output, \(s\) is the saving rate, \(n\) is the population growth rate, \(g\) is the output's growth rate, and \(\delta\) is the depreciation rate of capital. These examples highlight the relevance and importance of understanding the concepts, techniques, and applications of first-order differential equations in various fields. This knowledge serves as a solid foundation for further exploration and analysis of more advanced topics and real-world problems.