Question

True or False? In Exercises 67 and 68 , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. \(y^{\prime}+x y=e^{x} y\) is a first-order linear differential equation.

Short answer

The given equation \(y^{\prime} + xy = e^{x}y\) is not a linear first-order differential equation. Therefore, the statement is false.

Step-by-step solution

Identify the given equation

Here, the given equation is \(y^{\prime} + xy = e^{x}y\). It's important to discern whether it matches the standard form of a linear first-order differential equation, which is \(\frac{dy}{dx} + p(x)y = g(x)\).

Compare the structure of the given equation with the standard form

In the standard form, \(p(x)\) is a function that multiplies \(y\) and \(g(x)\) is a function that is on the right side of the equation. Rewriting the given equation in a similar form, we have \(y^{\prime} + xy - e^{x}y = 0\), which does not align perfectly with the standard form because the term \(e^{x}y\) on the left cannot simply be treated as a function of \(x\).

Make a conclusion

Because the term \(e^{x}y\) in the equation does not correspond to the standard form \(\frac{dy}{dx} + p(x)y = g(x)\), it can be concluded that the original statement is false. The given equation is not a linear first-order differential equation.

Key concepts

Differential Equations

Differential equations are mathematical tools used to describe various phenomena in the sciences, engineering, and beyond. They consist of equations involving derivatives which represent the rate of change of physical quantities. The process of finding a function when its derivative is given is called solving a differential equation.

These mathematical expressions can be both simple and highly complex, depending on the order of the derivative, linearity, and other contributing factors. They serve as the foundation for modeling dynamic systems such as population growth, electrical circuits, chemical reactions and even the motion of celestial bodies.

Linear Differential Equation

A linear differential equation is an equation that involves the derivatives of a function and can be written so no derivatives are multiplied together, and the function and its derivatives are only multiplied by functions of the independent variable.

The general first order linear differential equation can be expressed in the form \(\frac{dy}{dx} + p(x)y = g(x)\), where \(p(x)\) and \(g(x)\) are functions of x. It is termed 'linear' because the function y and its first derivative \(y'\) appear to the first power and are not multiplied together. The linearity implies that solutions to the equation can be uniquely determined and also have the 'superposition principle' applied to them.

Standard Form of Differential Equation

The standard form of a differential equation is essential for identifying the type of equation and the methods available for solving it. For a first-order linear differential equation, we often prefer the form \(\frac{dy}{dx} + p(x)y = g(x)\), where the derivative of y with respect to x is written on the left side of the equation, accompanied by a function \(p(x)\) times y equal to another function \(g(x)\) on the right side.

This standard form facilitates the use of certain techniques like the Integrating Factor method to solve the equation. It's crucial in ensuring the application of consistent methods across various problems and can greatly simplify the process of finding a solution.

Solving Differential Equations

Solving differential equations is the process of finding a function that satisfies the given equation. There are many techniques for solving differential equations, depending on their type and complexity. For first-order linear differential equations, common methods include Separation of Variables and Integrating Factors.

When presented with a first-order linear differential equation, the steps typically involve rearranging the terms into the standard form, identifying \(p(x)\) and \(g(x)\), and then implementing the appropriate solving technique. One critical aspect of this process is the initial condition, which is a value of the function at a particular point used to tailor the general solution to a specific case.