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Chapter 6: Problem 76

Determine whether each differential equation is separable. $$ y^{\prime}=x+y $$

Short Answer

Expert verified
The given differential equation is not separable.

Step by step solution

01

Analyze the Differential Equation

The given differential equation is \( y' = x + y \). We need to determine if it can be separated into two integrals: one in terms of \( x \) and one in terms of \( y \).
02

Rearrange the Equation

Rearrange the equation to separate terms involving \( y \) and terms involving \( x \). Subtract \( y \) from both sides to get \( y' - y = x \).
03

Determine Separability

For a differential equation to be separable, it can typically be written in the form \( g(y) \, dy = h(x) \, dx \). In the current rearrangement, \( y' = x + y \) or the equivalent \( y' - y = x \), we cannot easily separate variables because the term \( y' \) depends on both \( y \) and \( x \) without a clear multiplicative separation of variables.

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Key Concepts

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

Differential Equation
A differential equation is an equation that involves an unknown function and its derivatives. These equations are fundamental in explaining various phenomena in engineering, physics, economics, and other disciplines. For example, they are used to model the growth of populations or the cooling of an object. In this exercise, the differential equation presented is \( y' = x + y \).
This particular equation involves the first derivative of \( y \), denoted \( y' \), which is a common occurrence in what is known as a "first-order differential equation." Key Points:
  • Differential equations express relationships between quantities and the rates at which these quantities change.
  • Understanding how to manipulate differential equations helps in predicting behavior in systems.
  • First-order differential equations involve the first derivative of the unknown function.
Variable Separation
The method of variable separation is a technique used to solve some differential equations. The idea is to rewrite the equation so that all terms involving one variable are on one side, while all terms involving another variable are on the other. For variables like \( x \) and \( y \), you want to rearrange the equation to get terms featuring \( x \) with \( dx \) and \( y \) with \( dy \).In our exercise, we attempt to separate \( y' = x + y \) into terms of \( y \) and terms of \( x \). Unfortunately, separation is challenged by phrase-like structures such as "\( y' = x + y \)", where terms intermingle.Highlights:
  • Variable separation requires distinct groups of variable terms on each side of the equation.
  • Successfully separating the variables makes it easier to integrate each side independently.
  • Not all differential equations can be separated; recognizing this is crucial.
Rearranging Equations
Rearranging equations is a fundamental skill in tackling differential equations like \( y' = x + y \). The goal is to manipulate the equation, hoping to form a separable structure. The first step is to isolate terms by moving them between sides through basic arithmetic operations. When solving, pay attention to ensure linear terms involving the same variable are handled properly.Here's what we did:
  • Subtracted \( y \) from both sides to get \( y' - y = x \).
  • Attempted separation but faced a roadblock due to the interdependency of terms.
By rearranging equations effectively, we lay the groundwork for further analysis. However, remember that successful separation isn't always guaranteed, pointing the need for alternative methods.

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Most popular questions from this chapter

$$ \text { Use integration by parts to find each integral. } $$ $$ \int \sqrt[3]{x} \ln x d x $$

$$ \text { Use integration by parts to find each integral. } $$ $$ \int x^{5} \ln x d x $$

Derive each formula by using integration by parts on the left-hand side. (Assume \(n>0 .\) ) $$ \int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x $$

An annuity is a fund into which one makes equal payments at regular intervals. If the fund earns interest at rate \(r\) compounded continuously, and deposits are made continuously at the rate of \(d\) dollars per year (a "continuous annuity"), then the value \(y(t)\) of the fund after \(t\) years satisfies the differential equation \(y^{\prime}=d+r y\). (Do you see why?) Solve the differential equation above for the continuous annuity \(y(t)\), where \(d\) and \(r\) are unknown constants, subject to the initial condition \(y(0)=0\) (zero initial value).

Each equation follows from the integration by parts formula by replacing \(u\) by \(f(x)\) and \(v\) by a particular function. What is the function \(v\) ? $$ \int f(x) \frac{1}{x} d x=f(x) \ln x-\int \ln x f^{\prime}(x) d x $$
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