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Chapter 4: Problem 213

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

Short Answer

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

Step by step solution

01

Identify the Equation

The given differential equation is \( y' = x^3 y + \sin x \). Our goal is to write this equation in standard form.
02

Recall Standard Form of a First-Order DE

For a first-order linear differential equation, the standard form is \( y' + P(x)y = Q(x) \), where \( P(x) \) and \( Q(x) \) are functions of \( x \).
03

Rearrange to Match Standard Form

In the equation \( y' = x^3 y + \sin x \), rearrange it to match the form \( y' + P(x)y = Q(x) \). Move the \( x^3 y \) term to the left-hand side: \( y' - x^3 y = \sin x \).
04

Identify the Standard Form Components

Now that the equation \( y' - x^3 y = \sin x \) is in standard form, identify \( P(x) \) and \( Q(x) \). Here, \( P(x) = -x^3 \) and \( Q(x) = \sin x \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
The standard form of a first-order differential equation is essential for solving and analyzing these equations. In mathematics, a first-order linear differential equation can be expressed as \[ y' + P(x)y = Q(x) \]. This form allows us to see the structure of the equation more clearly. Here, \( y' \) represents the derivative of \( y \) with respect to \( x \).
  • \( P(x) \) is known as the coefficient of \( y \), which is a function of \( x \).
  • \( Q(x) \) is the function on the right-hand side that does not contain the dependent variable \( y \).
Understanding this setup is crucial for techniques such as integrating factors and other solution methods for linear differential equations. By writing an equation in this form, you lay the groundwork for finding explicit solutions or understanding the behavior of the solutions.
Linear Differential Equation
Linear differential equations, as the name suggests, are equations that involve a linear combination of a function and its derivatives. They play a crucial role in both mathematics and physics. For first-order linear differential equations, the general form is \[ y' + P(x)y = Q(x) \].They are considered 'linear' because:
  • The dependent variable \( y \) and its derivative \( y' \) appear to the first power, and are not multiplied together.
  • There are no products of the dependent variable and its powers or products of the derivative with itself.
By working within the framework of linear differential equations, it simplifies the process of finding analytical solutions and understanding the system鈥檚 behavior. These equations can often be solved using straightforward techniques, making them a staple in mathematical modeling.
Rearrange Equations
When dealing with first-order differential equations, rearranging terms to fit the standard form is a key step. Let's look at our original equation: \[ y' = x^3 y + \sin x \].To write this in standard form, we need to adjust the equation to match the structure \[ y' + P(x)y = Q(x) \].Here's how we do it:
  • Move all terms involving the dependent variable \( y \) to one side of the equation.
  • In our case, subtract \( x^3 y \) from both sides to isolate the terms involving \( y \): \[ y' - x^3 y = \sin x \].
This arrangement is crucial as it allows us to identify the necessary components \( P(x) \) and \( Q(x) \), forming a clear path to solving the equation.
Identify Components
Identifying the components of a first-order differential equation, once it is in standard form, is fundamental for solving it correctly. In our standard form\[ y' + P(x)y = Q(x) \],the terms \( P(x) \) and \( Q(x) \) must be clearly understood:
  • \( P(x) \) is the function that multiplies the dependent variable \( y \). In the equation: \[ y' - x^3 y = \sin x \], \( P(x) \) is identified as \(-x^3 \).
  • \( Q(x) \) stands for the part of the equation that does not involve \( y \). Here, it is \( \sin x \).
Recognizing these components lets you apply the appropriate solution techniques, such as integrating factors, effectively leading to solutions that describe the system's behavior or providing insights that might not be obvious initially.

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Most popular questions from this chapter

Draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. \(y^{\prime}=\frac{1}{x}+\ln x-y,\) for \(x>0\)

Use Euler's Method with \(n=5\) steps over the interval \(t=[0,1]\). Then solve the initial-value problem exactly. How close is your Euler's Method estimate? $$ y^{\prime}=-4 y x, y(0)=1 $$

163\. Solve the generic problem \(y^{\prime}=a y+b\) with initial condition \(y(0)=c\).

Solve the following initial-value problems by using integrating factors. $$ (2+x) y^{\prime}=y+2+x, y(0)=0 $$

Consider the logistic equation in the form \(P^{\prime}=C P-P^{2}\). Draw the directional field and find the stability of the equilibria. $$ C=3 $$
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