Problem 9 Solve difierential equations by ... [FREE SOLUTION]

Chapter 10: Problem 9

Solve difierential equations by series. From the equation \(d y / d x=y-2\) find all the derivatives of \(y\) at \(x=0\) starting from \(y(0)=1\). Construct the infinite series for \(y\), identify it as a known function, and verify that the function satisfies \(y^{\prime}=y-2\).

Short Answer

Expert verified

The solution is \( y(x) = 2 - e^x \).

Step by step solution

Derive the Series Expansion

We seek a series solution for the differential equation \( \frac{dy}{dx} = y - 2 \) about \( x = 0 \). Assume \( y(x) = \sum_{n=0}^{\infty} a_n x^n \). Given \( y(0) = 1 \), it follows that \( a_0 = 1 \).

Determine the First Derivative

First, find \( y'(x) \) by differentiating the series term by term: \( y'(x) = \sum_{n=1}^{\infty} n a_n x^{n-1} \). Setting \( y'(0) = y(0) - 2 \), we have \( n = 1 \) gives \( a_1 = y'(0) = 1 - 2 = -1 \).

Find Higher Derivatives

Continue differentiating to find higher order coefficients using the equation. For \( n = 2, 3, \dots \), equate the coefficients of the series \( y'(x) = y(x) - 2 \) term by term to find \( a_n \).

Calculate the Next Coefficient

For \( n = 2 \), equate to get \( 2a_2 = a_1 \rightarrow a_2 = \frac{-1}{2!} \). Repeat this for higher \( n \).

Construct and Identify the Infinite Series

The series that emerges is \( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots \), identified as the series for \( 2 - e^x \).

Verify the Function

Verify that \( y = 2 - e^x \) satisfies the differential equation. Calculate \( y' = -e^x \) and check that \( y' = y - 2 \rightarrow -e^x = (2 - e^x) - 2 \), thus verifying the solution works.

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Key Concepts

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

Series Expansion

To solve the differential equation \( \frac{dy}{dx} = y - 2 \) using a series expansion, we start by assuming that \( y(x) \) can be expressed as an infinite series. The series format looks like this:

\( y(x) = \sum_{n=0}^{\infty} a_n x^n \)

Using this representation, we can find expressions for the derivatives of \( y \) by differentiating the series term by term. This powerful method allows us to systematically find the coefficients \( a_n \) by equating them with terms from the right side of the differential equation. Starting from an initial condition such as \( y(0) = 1 \), giving that \( a_0 = 1 \), we proceed to find higher-order derivatives by matching terms.

Infinite Series

An infinite series is a series that continues indefinitely. In our problem, we find an infinite series representation of \( y \) as:

\( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \cdots \)

Each term in the series becomes smaller as \( n \) increases, a characteristic behavior of well-behaved infinite series. This series can continue without end, approximating a function very closely. Infinite series play a critical role in many areas of mathematics, particularly in finding solutions to equations that cannot be solved explicitly. Identifying the series as a known function simplifies this process considerably.

Known Function Identification

After deriving the series expansion, the next task is to recognize it as a familiar function. Recognizing familiar series patterns helps us translate series into well-known functions, simplifying calculations and more importantly verifying the solution. The series

\( 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!} + \dots \)

resembles the series expansion of \( e^x \), which is \( 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots \). By comparing, we identify our series corresponds to \( 2 - e^x \). Identifying such patterns makes the function easier to work with, especially for checking its validity against the original differential equation.

Verification of Solutions

Verification ensures that the obtained function satisfies the given differential equation. For the function \( y = 2 - e^x \), we need to check if it fulfills the requirement that \( y' = y - 2 \). Calculating the derivative,

\( y' = -e^x \)

We substitute back into the original equation to verify that:

\( -e^x = (2 - e^x) - 2 \)

After simplifying, we confirm that both sides equal \( -e^x \), demonstrating that the solution holds true. Verification is a crucial step in validating that the mathematical operations performed lead to an applicable and correct result.

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91影视

Solve difierential equations by series. From the equation \(d y / d x=y-2\) find all the derivatives of \(y\) at \(x=0\) starting from \(y(0)=1\). Construct the infinite series for \(y\), identify it as a known function, and verify that the function satisfies \(y^{\prime}=y-2\).

Short Answer

Step by step solution

Derive the Series Expansion

Determine the First Derivative

Find Higher Derivatives

Calculate the Next Coefficient

Construct and Identify the Infinite Series

Verify the Function

Key Concepts

Series Expansion

Infinite Series

Known Function Identification

Verification of Solutions

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