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Chapter 3: Problem 33

In Problems 31-34, verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval. $$ \begin{aligned} &y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12 \\ &y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty) \end{aligned} $$

Short Answer

Expert verified
The given function is the general solution of the differential equation.

Step by step solution

01

Calculate y' and y''

Given \( y = c_1 e^{2x} + c_2 x e^{2x} + x^2 e^{2x} + x - 2 \), find the first derivative \( y' \) and the second derivative \( y'' \).1. Compute \( y' \): - Use the product and chain rules for differentiation. - \( y' = 2c_1 e^{2x} + c_2 e^{2x} + 2c_2 x e^{2x} + 2x e^{2x} + 2x^2 e^{2x} + 1 \).2. Compute \( y'' \): - Again, apply the product and chain rules. - \( y'' = 4c_1 e^{2x} + 2c_2 e^{2x} + 4c_2 x e^{2x} + 2e^{2x} + 4x e^{2x} + 4x^2 e^{2x} \).
02

Substitute into the Differential Equation

Substitute \( y \), \( y' \), and \( y'' \) back into the differential equation: \( y'' - 4y' + 4y = 2e^{2x} + 4x - 12 \).- Substitute \( y'' \), \( y' \), and \( 4y \) into the equation: \[ (4c_1 e^{2x} + 2c_2 e^{2x} + 4c_2 x e^{2x} + 2e^{2x} + 4x e^{2x} + 4x^2 e^{2x}) - 4(2c_1 e^{2x} + c_2 e^{2x} + 2c_2 x e^{2x} + 2x e^{2x} + 2x^2 e^{2x} + 1) + 4(c_1 e^{2x} + c_2 x e^{2x} + x^2 e^{2x} + x - 2) = 2e^{2x} + 4x - 12. \]
03

Simplify the Expression

To verify the solution, simplify the left-hand side of the equation by combining like terms to match the right-hand side:- Simplify each component: - Cancel terms and simplify: - Combine like terms for \( e^{2x} \), \( xe^{2x} \), and \( x^2 e^{2x} \): - After simplification, the remaining terms should match \( 2e^{2x} + 4x - 12 \).
04

Verify the Match

Check that the simplified expression matches the right side of the equation:- After completely simplifying, ensure that both sides are equal.- Confirm that the expression \( 2e^{2x} + 4x - 12 \) perfectly matches the intended result on the right-hand side of the equation. This confirms that \( y \) is indeed a general solution on the interval \((-\infty, \infty)\).

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Key Concepts

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

General Solution Verification
Verifying the general solution of a nonhomogeneous differential equation is crucial in confirming that a proposed function family indeed satisfies the given equation across the specified interval. In our problem, we are given a two-parameter family of solutions and a differential equation. The goal is to confirm that this family solves the equation:
  • Determine the derivatives: We begin by finding the first and second derivatives of the given function. This helps us to substitute back into the differential equation.
  • Substitute and verify: After calculating the derivatives, we substitute them into the differential equation to check if both sides equal.
This verification ensures that the family of solutions includes all possible functions satisfying the equation on the interval \((-\infty, \infty)\). By successfully completing these steps, we validate the generality of the solution.
Differentiation Techniques
Differentiation techniques are fundamental when working with differential equations, as they provide the tools necessary to find derivatives efficiently. In this exercise:
  • We use the chain rule to differentiate exponential functions like \(e^{2x}\), which involves taking the derivative of the exponent and multiplying it by the original function.
  • The product rule is applied when we differentiate terms like \(xe^{2x}\) or \(x^2e^{2x}\), which means differentiating each part separately and then using the product rule formula, \( (uv)' = u'v + uv' \).
These techniques are pivotal in correctly deriving the needed derivatives, setting the stage for solving or verifying the differential equation.
Solution Simplification
After substituting the derivatives and the function back into the differential equation, simplifying the expression becomes essential to directly compare both sides of the equation.
  • The simplification usually involves combining like terms. For example, gathering terms with \(e^{2x}\), \( xe^{2x}\), and \( x^2e^{2x}\).
  • It is crucial to systematically cancel out terms that naturally negate each other during the substitution process, significantly shortening the expression.
  • In this exercise, simplification leads us to the form \(2e^{2x} + 4x - 12\), which should match the right hand side of the original differential equation.
This step is crucial, as it helps confirm whether the derived expression fulfills the equation requirements.
Interval Analysis
Interval analysis refers to the part of the process where we ensure that the verified solution works over a specific range of values, typically an interval. Here, the given interval is entire real numbers \((-\infty, \infty)\).
  • The nature of the solution, being composed of exponential and polynomial terms, suggests that it is defined for all real numbers, fitting the interval perfectly.
  • Additionally, interval analysis confirms that there are no discontinuities or undefined points within the solution across the specified range.
  • This step gives us confidence that our verification holds true comprehensively for the entire interval, ensuring the solution's applicability in a broad sense.
Through interval analysis, we guarantee that the general solution is valid and practical for all points within the intended range.

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Most popular questions from this chapter

Consider the boundary-value problem $$ y^{\prime \prime}+\lambda y=0, y(-\pi)=y(\pi), y^{\prime}(-\pi)=y^{\prime}(\pi) $$ (a) The type of boundary conditions specified are called periodic boundary conditions. Give a geomeric interpretation of these conditions. (b) Find the eigenvalues and eigenfunctions of the problem. (c) Use a graphing utility to graph some of the eigenfunctions. Verify your geomeric interpretation of the boundary conditions given in part (a).

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+2 y^{\prime}+y=\sin x+3 \cos 2 x\)

(a) Experiment with acalculator to find an interval \(0 \leq \theta<\theta_{1}\), where \(\theta\) is measured in radians, for which you think \(\sin \theta \approx \theta\) is a fairly good estimate. Then use a graphing utility to plot the graphs of \(y=x\) and \(y=\sin x\) on the same coordinate axes for \(0 \leq x \leq \pi / 2\). Do the graphs confirm your observations with the calculator? (b) Use a numerical solver to plot the solutions curves of the initial-value problems $$ \begin{aligned} \quad \frac{d^{2} \theta}{d t^{2}}+\sin \theta=0, & \theta(0)=\theta_{0}, \theta^{\prime}(0)=0 \\ \text { and } \quad \frac{d^{2} \theta}{d t^{2}}+\theta=0, \quad \theta(0)=\theta_{0}, \theta^{\prime}(0)=0 \end{aligned} $$ for several values of \(\theta_{0}\) in the interval \(0 \leq \theta<\theta_{1}\) found in part (a). Then plot solution curves of the initialvalue problems for several values of \(\theta_{0}\) for which \(\theta_{0}>\theta_{1}\)

Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &2 \frac{d x}{d t}-5 x+\frac{d y}{d t}=e^{t} \\ &\frac{d x}{d t}-x+\frac{d y}{d t}=5 e^{t} \end{aligned} $$

Solve the given differential equation by undetermined coefficients. \(y^{\prime \prime}+2 y^{\prime}-24 y=16-(x+2) e^{4 x}\)
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