Question

Write the following first-order differential equations in standard form. $$ y^{\prime}+3 y-\ln x=0 $$

Short answer

The equation in standard form is \( y' + 3y = \ln x \).

Step-by-step solution

Identify the Given Equation

The exercise provides us with the first-order differential equation: \( y' + 3y - \ln x = 0 \). Our objective is to determine if it is already in the standard form or convert it if necessary.

Determine the Standard Form

The standard form for a linear first-order differential equation is \( y' + P(x)y = Q(x) \). In this form, \( y' \) is isolated, and the equation is structured with \( P(x) \) being a function of \( x \) that multiplies \( y \), and \( Q(x) \) being a function of \( x \) on the other side of the equation.

Rearrange the Equation into Standard Form

The given equation is \( y' + 3y - \ln x = 0 \). We can rearrange it to fit the standard form by moving \( \ln x \) to the other side. This yields \( y' + 3y = \ln x \), which now matches our desired structure \( y' + P(x)y = Q(x) \) where \( P(x) = 3 \) and \( Q(x) = \ln x \).

Key concepts

Linear Differential Equations

Linear differential equations are a specific type of differential equations that form the backbone of many scientific and engineering problems. They feature the unknown function and its derivatives, and the equation is linear in the unknown function and its derivatives.
Linear refers to the idea that the equation can be written in such a way that no powers or products of the unknown function and its derivatives appear in the equation. This leads to a structure that can often be solved more straightforwardly.
A key characteristic of these equations is their order, which is determined by the highest derivative present. First-order linear differential equations are a common type, meaning they involve only the first derivative of the unknown function.
In general, a linear differential equation can be expressed in the form \( a_n(x) y^{(n)} + a_{n-1}(x) y^{(n-1)} + ... + a_1(x) y' + a_0(x) y = g(x) \), but for first-order, this simplifies considerably.

• An example of a first-order linear differential equation is \( y' + a(x)y = g(x) \).

Understanding the linear nature of these equations helps in applying techniques that leverage this simplicity to find solutions.

Standard Form

The standard form of a first-order linear differential equation makes it organized and easier to identify key parts of the equation. It is described by the formula \( y' + P(x) y = Q(x) \).
Rewriting an equation to this form aligns with solving practices and allows for a straightforward recognition of what is changing in the equation. Importantly, this form helps to identify the coefficient functions directly.
Let's break down the components of the standard form:

• \( y' \): The derivative of the function \( y \), indicating the rate of change.
• \( P(x) \): A function of \( x \) that is multiplied by \( y \), serving as a linear coefficient. In many text problems, \( P(x) \) is often a constant or a simple function of \( x \).
• \( Q(x) \): Another function of \( x \), found on the right side of the equation. It is not multiplied by \( y \) or its derivatives, representing the independent variation.

By organizing equations in this way, solving techniques such as the integrating factor method become far easier to implement, greatly simplifying the process of finding solutions.

Differential Equations Solution

Finding the solution to a differential equation means identifying the function that satisfies the equation for all points in the domain. For linear first-order differential equations, this usually involves the integration of functions.
When dealing with the standard form \( y' + P(x)y = Q(x) \), one common method for finding solutions is using an integrating factor. This factor allows the differential equation to be written in a way that enables straightforward integration.
Here is how the process generally works:

• Calculate the integrating factor, \( \mu(x) = e^{\int P(x) \, dx} \).
• Multiply the entire differential equation by this integrating factor.
• The left side of the equation becomes the derivative of a product, \( y \mu(x) \), allowing us to integrate both sides easily.
• Solve for \( y(x) \) by performing the integrations and simplifying.

The beauty of this method lies in transforming a potentially complex equation into one that integrates neatly, leading to a general solution that can explain a wide variety of phenomena in mathematical modeling and real-world applications.