Problem 16 Write a trial solution for the m... [FREE SOLUTION]

Chapter 17: Problem 16

Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $$ y^{\prime \prime}+3 y^{\prime}-4 y=\left(x^{3}+x\right) e^{x} $$

Short Answer

Expert verified

Trial solution: $(Ax^4 + Bx^3 + Cx^2 + Dx)e^x$.

Step by step solution

Identify the Type of Non-Homogeneous Term

The non-homogeneous term on the right-hand side of the differential equation is $(x^3 + x)e^x$. This is a product of a polynomial $(x^3 + x)$ and an exponential function $e^x$.

Determine the Form of the Trial Solution

Since the non-homogeneous term $(x^3 + x)e^x$ is a combination of a polynomial and an exponential function, the trial solution will need to account for both. The general form of the trial solution will take the form $(Ax^3 + Bx^2 + Cx + D)e^x$, where $A$, $B$, $C$, and $D$ are coefficients to be determined.

Check for Overlap with Homogeneous Solution

Check the characteristic equation of the homogeneous part $y'' + 3y' - 4y = 0$ to ensure the trial solution does not overlap. The characteristic equation is $r^2 + 3r - 4 = 0$, which gives roots $r = 1$ and $r = -4$. The root $r = 1$ indicates the solution of the homogeneous equation includes terms with $e^x$. Thus, the trial solution must be multiplied by $x$ to ensure it's not a part of the homogeneous solution, resulting in $(Ax^4 + Bx^3 + Cx^2 + Dx)e^x$.

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Key Concepts

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

Second Order Linear Differential Equations

Second order linear differential equations are a crucial part of calculus and mathematical modeling. These equations include the second derivative of a function, typically represented as $ y'' $. The standard form of a second order linear differential equation is:

$ a y'' + b y' + c y = g(x) $

where $ y'' $ is the second derivative, $ y' $ is the first derivative, and $ g(x) $ represents a non-homogeneous term.
This form of equations helps model systems with forces or accelerations, like spring-mass systems or circuits. Recognizing the order and terms in these equations sets the stage for applying methods like undetermined coefficients.

Non-Homogeneous Differential Equations

A non-homogeneous differential equation includes a non-zero function on the right side, often denoted as $ g(x) $. This term makes the equation unique and more complex than its homogeneous counterpart. In our example, the non-homogeneous term is given by:

$ (x^3 + x)e^x $

The non-homogeneous term dictates the trial solution's structure. It often includes polynomials, exponentials, sines, or cosines. The method of undetermined coefficients uses the form of this term to craft a trial solution that accounts for these extra components.

Characteristic Equation

The characteristic equation is a key step in solving linear differential equations. For the homogeneous part, it is derived by assuming a solution of the form $ y = e^{rx} $ and substituting into the differential equation. This results in an algebraic equation. For example, the characteristic equation for:

$ y'' + 3y' - 4y = 0 $

is:

$ r^2 + 3r - 4 = 0 $

Solving this equation provides the roots, $ r $, which indicate the nature of the homogeneous solution. Here, the roots are $ r = 1 $ and $ r = -4 $, leading to solutions containing $ e^x $ and $ e^{-4x} $. Knowing these roots is essential in crafting a trial solution for the non-homogeneous part without redundancy.

Trial Solution

The trial solution is a formative step in the method of undetermined coefficients. It approximates the solution based on the form of the non-homogeneous term. In our example, the initial trial solution might look like:

$ (Ax^3 + Bx^2 + Cx + D)e^x $

This accounts for the polynomial and exponential part of the non-homogeneous term. However, because $ e^x $ is a solution for the homogeneous equation, the trial solution must be modified by multiplying by $ x $ to prevent duplication. Thus, the adjusted trial solution becomes:

$ (Ax^4 + Bx^3 + Cx^2 + Dx)e^x $

This ensures that it is linearly independent of the homogeneous solutions.

Homogeneous Solution

The homogeneous solution refers to the solution of the differential equation when the non-homogeneous term $ g(x) $ is zero. Using our equation's characteristic roots $ r = 1 $ and $ r = -4 $, the general form of the homogeneous solution is:

$ y_h = C_1 e^x + C_2 e^{-4x} $

where $ C_1 $ and $ C_2 $ are constants determined by initial conditions. This solution captures the behavior of the system independent of external forces or inputs. It serves as a foundation upon which the particular solution, from the trial solution, builds to solve the entire non-homogeneous equation.

91影视

Write a trial solution for the method of undetermined coefficients. Do not determine the coefficients. $$ y^{\prime \prime}+3 y^{\prime}-4 y=\left(x^{3}+x\right) e^{x} $$

Short Answer

Step by step solution

Identify the Type of Non-Homogeneous Term

Determine the Form of the Trial Solution

Check for Overlap with Homogeneous Solution

Key Concepts

Second Order Linear Differential Equations

Non-Homogeneous Differential Equations

Characteristic Equation

Trial Solution

Homogeneous Solution

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