Problem 15 Solve the differential equation.... [FREE SOLUTION]

Chapter 19: Problem 15

Solve the differential equation. \(y^{\prime \prime}-8 y^{\prime}+16 y=0\)

Short Answer

Expert verified

The general solution is \(y(x) = C_1 e^{4x} + C_2 x e^{4x}\).

Step by step solution

Identify the Type of Differential Equation

The given differential equation is a second-order linear homogeneous differential equation with constant coefficients. It's written in the standard form: \(y'' - 8y' + 16y = 0\).

Set Up the Characteristic Equation

To solve this type of differential equation, we find the characteristic equation, which is formed by replacing \(y''\) with \(r^2\), \(y'\) with \(r\), and \(y\) with 1. This gives us the characteristic equation: \(r^2 - 8r + 16 = 0\).

Solve the Characteristic Equation

Solve for \(r\) by factoring or using the quadratic formula. The characteristic equation \(r^2 - 8r + 16 = 0\) can be factored as \((r - 4)^2 = 0\). Thus, \(r = 4\) is a repeated root of multiplicity 2.

Write the General Solution

Since we have a repeated root \(r = 4\), the general solution for the differential equation is given by:\[y(x) = C_1 e^{4x} + C_2 x e^{4x}\]where \(C_1\) and \(C_2\) are arbitrary constants.

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Key Concepts

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

homogeneous differential equation

A homogeneous differential equation is a type of differential equation where the result is zero. In particular, we have equations of the form:

Second-order linear homogeneous equations with constant coefficients, like \(y'' - 8y' + 16y = 0\).

In these equations, all terms contain the function \(y\) or its derivatives. There is no independent term without \(y\). The focus is on the relationship between the function and its derivatives.
Homogeneous equations are crucial because they often model natural phenomena where things return to equilibrium over time. Understanding them helps solve more complex systems or non-homogeneous equations by using superposition principles.

characteristic equation

The characteristic equation is a tool used to solve linear differential equations with constant coefficients. It turns the differential equation into an algebraic one by substituting derivatives with powers of a variable, typically \(r\).
For the given equation \(y'' - 8y' + 16y = 0\), the characteristic equation is:

Substitute \(y''\), \(y'\), and \(y\) with \(r^2\), \(r\), and \(1\) respectively.
This yields: \(r^2 - 8r + 16 = 0\).

Once the characteristic equation is formed, solving it provides the roots, which indicate the type of solutions. These roots help craft a general solution for the differential equation based on their nature (real, imaginary, or repeated).

repeated roots

Repeated roots occur when the characteristic equation has the same solution more than once. In our example, solving \(r^2 - 8r + 16 = 0\) yields a repeated root \(r = 4\) with multiplicity two.
Repeated roots require a modified approach to form the general solution. When the characteristic equation has a repeated root \(r\), the general solution gains an additional term involving \(x\):

The solution takes the form \(y(x) = C_1 e^{rx} + C_2 x e^{rx}\).

This format accounts for the separate yet interconnected behaviors of each part of the solution. It ensures every possible solution type is included in the overall general solution.

general solution

The general solution combines all possible solutions of a differential equation derived from the roots of its characteristic equation. Our differential equation, \(y'' - 8y' + 16y = 0\), features repeated roots, leading us to this general solution:

\(y(x) = C_1 e^{4x} + C_2 x e^{4x}\)

Here, \(C_1\) and \(C_2\) are arbitrary constants. These constants ensure flexibility, allowing for all specific solutions depending on initial or boundary conditions.
The general solution is important as it summarizes all potential behaviors of a system. It offers insights into how a system responds to different initial inputs or perturbations, ensuring comprehensiveness in analysis and prediction.

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91影视

Solve the differential equation. \(y^{\prime \prime}-8 y^{\prime}+16 y=0\)

Short Answer

Step by step solution

Identify the Type of Differential Equation

Set Up the Characteristic Equation

Solve the Characteristic Equation

Write the General Solution

Key Concepts

homogeneous differential equation

characteristic equation

repeated roots

general solution

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