Question

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

Short answer

The standard form is: \( y' + (-x^3)y = \sin x \).

Step-by-step solution

Identifying Components

First, let's identify the components in the given differential equation. The equation provided is \( y' = x^3 y + \sin x \). Here, we have \( y' \) as the derivative term, \( x^3 y \) as a term involving \( y \), and \( \sin x \) as an inhomogeneous term.

Rearrange the Equation

To write the equation in standard form, we need to have all terms involving \( y \) on one side and all other terms on the other side. Start by subtracting \( x^3 y \) from both sides of the equation: \( y' - x^3 y = \sin x \).

Standard Form Identification

A first-order linear differential equation in standard form is given by \( y' + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are continuous functions of \( x \). Our equation is \( y' - x^3 y = \sin x \), which can be written as \( y' + (-x^3)y = \sin x \), identifying \( P(x) = -x^3 \) and \( Q(x) = \sin x \).

Key concepts

First-Order Differential Equation

A first-order differential equation involves derivatives of a function with respect to a single variable, typically denoted as \( y' \). These equations involve only the first derivative of the function, hence the name "first-order."

These equations can describe a wide range of phenomena in the physical and natural sciences, such as the rate at which a car slows down or how the population of an animal species changes over time. In the given exercise, the equation \( y' = x^3 y + \sin x \) is a classic example, involving a first derivative \( y' \).

To solve first-order differential equations, one often seeks to express the solution as a function \( y(x) \), which satisfies the equation within a specific domain of \( x \). Techniques like separation of variables, integrating factor method, or even numerical methods may be used to find solutions.

Standard Form

The standard form of a first-order linear differential equation is written as:
\ \[ y' + P(x)y = Q(x) \]

In this form, \( P(x) \) and \( Q(x) \) are functions of \( x \) that describe the behavior of the system modeled by the equation. The equation is rearranged so that the derivative term \( y' \) and a linear term involving \( y \) are on one side, with any additional terms shifted to the other side.

For the equation \( y' = x^3 y + \sin x \), it can be written in the standard form as:
\ \[ y' + (-x^3)y = \sin x \]

Here, \( P(x) = -x^3 \) and \( Q(x) = \sin x \). By positioning the equation in this standard form, it becomes easier to apply methods for finding a solution, especially when using the integrating factor method.

Inhomogeneous Term

The term \( \sin x \) in the equation \( y' + (-x^3)y = \sin x \) is referred to as the inhomogeneous term. An inhomogeneous term is any part of a differential equation that doesn't depend on the unknown function \( y \) or its derivatives, but rather only on the independent variable \( x \).

This means that the equation is not homogeneous, or uniform, in its treatment of \( y \). The presence of the inhomogeneous term \( Q(x) = \sin x \) implies that the solution will not be of the simple form as that of a homogeneous differential equation.

When working with such equations, the solution typically consists of two parts: a complementary (or homogeneous) solution that solves the associated homogeneous equation and a particular solution that accounts for the inhomogeneous term. The combination of these solutions gives the general solution to the differential equation.