Question

How do differential equations with constant coefficients differ from those with variable coefficients? Give an example for each type.

Short answer

Provide an example for each type. The main difference between differential equations with constant coefficients and those with variable coefficients is the nature of the coefficient term. In differential equations with constant coefficients, the coefficient term remains the same throughout the equation, while in those with variable coefficients, it changes with respect to the independent variable. Example with constant coefficients: Consider the differential equation: \(\frac{dy}{dx} + 2y = 4\) The general solution to this first-order linear differential equation with constant coefficients is: \(y(x) = 2 + \frac{C}{e^{2x}}\) Example with variable coefficients: Consider the differential equation: \(\frac{dy}{dx} + \frac{1}{x}y = x^2\) The general solution to this first-order linear differential equation with variable coefficients is: \(y(x) = \frac{1}{4}x^2 + \frac{C}{x^2}\) In summary, the main differences between these types of differential equations are both the form of the coefficient term and the methods used to solve them.

Step-by-step solution

Introduction

A differential equation of the form \(\frac{dy}{dx} + a(x)y = f(x)\) is called a first-order linear differential equation. When the term \(a(x)\) is a constant, then the equation has constant coefficients. If the term \(a(x)\) depends on \(x\), then the equation has variable coefficients.

Example with constant coefficients

Let's consider a first-order linear differential equation with constant coefficients: $$\frac{dy}{dx} + 2y = 4$$ To solve this equation, we can use an integrating factor, which is given by \(e^{\int a(x)dx}\). In this case, \(a(x) = 2\), so the integrating factor is: $$e^{\int 2 dx} = e^{2x}$$ Now, we can multiply the equation by the integrating factor: $$e^{2x} \frac{dy}{dx} + 2e^{2x}y = 4e^{2x}$$ Observe that the left-hand side of the equation is the derivative of the product \(e^{2x}y\). Thus, we can write: $$\frac{d}{dx}(e^{2x}y) = 4e^{2x}$$ Now, integrate both sides with respect to \(x\): $$\int \frac{d}{dx}(e^{2x}y) dx = \int 4e^{2x} dx$$ $$e^{2x}y = 2e^{2x} + C$$ Lastly, divide both sides by \(e^{2x}\): $$y(x) = 2 + \frac{C}{e^{2x}}$$ This is the general solution to our first-order linear differential equation with constant coefficients.

Example with variable coefficients

Now let's consider a first-order linear differential equation with variable coefficients: $$\frac{dy}{dx} + \frac{1}{x}y = x^2$$ The integrating factor in this case is: $$e^{\int \frac{1}{x} dx} = e^{\ln x} = x$$ Multiplying the equation by the integrating factor, we get: $$x \frac{dy}{dx} + xy = x^3$$ The left-hand side is the derivative of the product \(x^2y\). Thus, we can write: $$\frac{d}{dx}(x^2y) = x^3$$ Now, integrate both sides with respect to \(x\): $$\int \frac{d}{dx}(x^2y) dx = \int x^3 dx$$ $$x^2y = \frac{1}{4}x^4 + C$$ Lastly, divide both sides by \(x^2\): $$y(x) = \frac{1}{4}x^2 + \frac{C}{x^2}$$ This is the general solution to our first-order linear differential equation with variable coefficients.

Conclusion

The main difference between differential equations with constant coefficients and those with variable coefficients is the nature of the coefficient term. The constant coefficient remains the same throughout the equation, while the variable coefficient changes with respect to the independent variable. The examples provided demonstrate this difference in form and solution methods.

Key concepts

Constant Coefficients

Differential equations with constant coefficients are those where the terms influencing the rate of change are constant, meaning they do not depend on the independent variable, usually symbolized as \(x\). This creates simplicity in solving the equation as the coefficient remains unchanged regardless of the value of \(x\).
In the first-order linear differential equation form \(\frac{dy}{dx} + ay = f(x)\), if \(a\) is a constant, the equation has constant coefficients. For example, the equation \(\frac{dy}{dx} + 2y = 4\) has a constant coefficient of 2.
To solve these types of equations, you use an integrating factor, which, due to the constant nature, takes a predictable form. This is crucial because it helps to standardize the method of finding solutions, making calculations straightforward.
One of the most useful aspects of constant coefficients is that they simplify during calculations, allowing the use of standard algebraic techniques and reducing the complexity of integration needed to find the general solution.

Variable Coefficients

In contrast to constant coefficients, differential equations with variable coefficients involve terms that change contextually with the variable \(x\). This means the term that multiplies \(y\), typically represented as \(a(x)\), varies depending on the independent variable's value, complicating the solution process.
Consider a simple first-order linear differential equation of the form \(\frac{dy}{dx} + \frac{1}{x}y = x^2\). Here, the coefficient \(\frac{1}{x}\) changes as \(x\) changes. The varying nature requires a different approach and makes using an integrating factor more challenging.
The integrating factor for variable coefficients takes into account the changing nature of \(a(x)\), making it crucial to carefully calculate \(\int a(x) \, dx\). For instance, the integrating factor \(e^{\int \frac{1}{x} \, dx}\) simplifies to \(e^{\ln x}\), which equals \(x\).
Solutions to these equations often require diligent calculation and may involve more complex functions or special techniques to integrate and solve the resulting equations.

Integrating Factor

An integrating factor is a mathematical tool used to transform a differential equation into a format that is easier to solve. It is particularly useful for first-order linear differential equations. The beauty of the integrating factor is that it essentially "cleans up" these equations by modifying them to have common terms that can be easily integrated.
The standard approach involves multiplying the entire differential equation by a function \(\mu(x) = e^{\int a(x) \, dx}\), which depends on the form of \(a(x)\). This allows the left-hand side of the equation to become the derivative of a product of the integrating factor and the dependent variable \(y\).
Consider the equation \(\frac{dy}{dx} + a(x)y = f(x)\). By multiplying through by the integrating factor, it restructures the equation as \(\frac{d}{dx}[e^{\int a(x) \, dx} \cdot y] = e^{\int a(x) \, dx} \cdot f(x)\).
This adjustment is what enables straightforward integration of both sides, simplifying the process of finding a solution and highlighting the interconnected role of constants and variables in the coefficient.

First-Order Linear Differential Equation

First-order linear differential equations are among the simplest types of differential equations to analyze, yet they carry significant weight in both theoretical and applied mathematics. These equations take on the general form of \(\frac{dy}{dx} + a(x)y = f(x)\), making them linear with respect to the unknown function \(y\).
Such equations can model a wide range of real-world phenomena, from exponential growth and decay to cooling models and even electrical circuits.
The solution of first-order linear differential equations generally involves understanding whether coefficients are constant or variable. This distinction informs how you select integrating factors and follow through the solution process.
Understanding these equations often opens the door to more complex differential equations, serving as a fundamental building block in mathematics.

• Real-world application relevance, such as in physics and engineering
• Simplicity in mathematical theory due to the linear nature
• Foundation for higher-order differential equations and systems analysis