Book cover for Advanced Engineering Mathematics

Advanced Engineering Mathematics

Dennis G. Zill, Michael R. Cullen

ISBN #9780763740955

3rd Edition

4,310 Questions

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17,647 Students Helped

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section covers a wide range of methods for solving differential equations – from classical methods such as reduction of order, undetermined coefficients, and variation of parameters to specialized techniques for Cauchy–Euler equations. Additionally, it introduces boundary–value problems where eigenvalues and eigenfunctions must be determined, and the Laplace transform is defined and applied. Key insights include the importance of the Wronskian in ensuring solution independence and careful application of boundary conditions to obtain unique or nontrivial solutions. These techniques are critical in modeling physical systems such as oscillating springs, buckling beams, and electrical circuits.

Learning Objectives

1

Explain and solve higher?order linear differential equations using a variety of methods including reduction of order, undetermined coefficients, and variation of parameters.

2

Describe and solve Cauchy–Euler (equidimensional) equations and interpret eigenvalue problems and boundary–value problems.

3

Apply the Laplace transform to convert differential equations into algebraic equations, and use it to solve initial–value problems.

4

Analyze systems of linear differential equations and understand the role of the Wronskian in determining linear independence.

5

Model physical systems (e.g. spring–mass systems, buckling columns, electrical circuits) using linear and nonlinear differential equations.

Key Concepts

CONCEPT

DEFINITION

Homogeneous Differential Equation

An equation in which every term is a function of the unknown function and its derivatives – its right–hand side is zero.

Particular Solution

A specific solution that accounts for the nonhomogeneous (forcing) term; the general solution is the sum of the homogeneous solution and a particular solution.

Wronskian

A determinant computed from a set of solutions used to test linear independence of those solutions.

Undetermined Coefficients

A method for finding particular solutions when the forcing function is of a special form (exponentials, polynomials, sines, cosines).

Variation of Parameters

A method that uses the fundamental set of solutions to the associated homogeneous equation to build a particular solution for the nonhomogeneous case.

Cauchy–Euler Equation

An equidimensional differential equation in which coefficients are powers of the independent variable; solved by assuming a solution of the form x^m.

Laplace Transform

An integral transform that converts a function of time into a function of the complex variable s, defined by L{f(t)} = ∫₀∞ e^(–st) f(t) dt.

Eigenvalue Problem

A boundary–value problem that leads to nontrivial solutions only for specific parameter values (eigenvalues) with corresponding eigenfunctions.

Example Problems

Example 1

From $y=c_{1} e^{x}+c_{2} e^{-x}$ we find $y^{\prime}=c_{1} e^{x}-c_{2} e^{-x} .$ Then $y(0)=c_{1}+c_{2}=0, y^{\prime}(0)=c_{1}-c_{2}=1$ so that $c_{1}=\frac{1}{2}$ and $c_{2}=-\frac{1}{2} .$ The solution is $y=\frac{1}{2} e^{x}-\frac{1}{2} e^{-x}.$

Example 2

From $y=c_{1} e^{4 x}+c_{2} e^{-x}$ we find $y^{\prime}=4 c_{1} e^{4 x}-c_{2} e^{-x} .$ Then $y(0)=c_{1}+c_{2}=1, y^{\prime}(0)=4 c_{1}-c_{2}=2$ so that $c_{1}=\frac{3}{5}$ and $c_{2}=\frac{2}{5} .$ The solution is $y=\frac{3}{5} e^{4 x}+\frac{2}{5} e^{-x}.$

Example 3

From $y=c_{1} x+c_{2} x \ln x$ we find $y^{\prime}=c_{1}+c_{2}(1+\ln x) .$ Then $y(1)=c_{1}=3, y^{\prime}(1)=c_{1}+c_{2}=-1$ so that $c_{1}=3$ and $c_{2}=-4 .$ The solution is $y=3 x-4 x \ln x.$

Example 4

From $y=c_{1}+c_{2} \cos x+c_{3} \sin x$ we find $y^{\prime}=-c_{2} \sin x+c_{3} \cos x$ and $y^{\prime \prime}=-c_{2} \cos x-c_{3} \sin x .$ Then $y(\pi)=$ $c_{1}-c_{2}=0, y^{\prime}(\pi)=-c_{3}=2, y^{\prime \prime}(\pi)=c_{2}=-1$ so that $c_{1}=-1, c_{2}=-1,$ and $c_{3}=-2 .$ The solution is $y=-1-\cos x-2 \sin x.$

Example 5

From $y=c_{1}+c_{2} x^{2}$ we find $y^{\prime}=2 c_{2} x .$ Then $y(0)=c_{1}=0, y^{\prime}(0)=2 c_{2} \cdot 0=0$ and hence $y^{\prime}(0)=1$ is not possible. since $a_{2}(x)=x$ is 0 at $x=0,$ Theorem 3.1 is not violated.
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Step-by-Step Explanations

QUESTION

Solve y'' − y = 1 + e^x by finding a particular solution and then the general solution.

STEP-BY-STEP ANSWER:

Step 1: Note that the associated homogeneous equation is y'' − y = 0 with general solution y_c = c₁ e^x + c₂ e^(–x).
Step 2: Since the forcing term is 1 + e^x, propose a particular solution of the form y_p = A + B e^x.
Step 3: Differentiate: y_p' = B e^x and y_p'' = B e^x.
Step 4: Substitute into the differential equation: B e^x − (A + B e^x) = 1 + e^x.
Step 5: Simplify to get –A = 1 and (B − B) e^x = e^x, so A = –1 and the coefficient of e^x yields 0 = e^x, which indicates that we must adjust the guess.
Step 6: Since e^x is already a solution of the homogeneous equation, multiply the e^x part by x. Instead, try y_p = A + B x e^x.
Step 7: Compute derivatives: y_p' = B e^x + Bx e^x; y_p'' = 2B e^x + Bx e^x.
Step 8: Substitute into the ODE: (2B e^x + Bx e^x) − [A + B x e^x] = 1 + e^x, yielding 2B e^x − A = 1 + e^x.
Step 9: Equate coefficients: For the constant term, –A = 1 so A = –1; for the e^x term, 2B = 1 so B = 1/2.
Step 10: Thus, the particular solution is y_p = –1 + (1/2) x e^x, and the general solution is y = c₁ e^x + c₂ e^(–x) – 1 + (1/2) x e^x.
Final Answer:

Undetermined Coefficients (Example)

QUESTION

Given a Cauchy–Euler equation x^2 y'' − 4xy' + 6y = 0 with one solution y1 = x, find a second solution.

STEP-BY-STEP ANSWER:

Step 1: Assume a second solution of the form y = u(x)y₁(x) = u(x)x.
Step 2: Compute derivatives: y' = u'x + u and y'' = u'' x + 2u'.
Step 3: Substitute into the differential equation: x^2 (u'' x + 2u') − 4x (u'x + u) + 6(x u) = 0.
Step 4: Simplify to obtain x^3 u'' + 2x^2 u' − 4x^2 u' − 4x u + 6x u = 0, or x^3 u'' − 2x^2 u' + 2x u = 0.
Step 5: Divide through by x (x ≠ 0): x^2 u'' − 2x u' + 2u = 0.
Step 6: Recognize that this equation in u can be simplified by setting v = u'. Solve for u using an appropriate method (or note that by inspection a solution u that leads to a second solution exists).
Step 7: In many cases the second solution can be written as y₂ = x ln|x|. (For the given example, the definite answer might vary.)
Final Answer:

Reduction of Order (Cauchy–Euler Example)

QUESTION

Find the Laplace transform of f(t) = e^(–t) sin t.

STEP-BY-STEP ANSWER:

Step 1: Write the definition L{f(t)} = ∫₀∞ e^(–st) f(t) dt.
Step 2: Substitute f(t): L{e^(–t) sin t} = ∫₀∞ e^(–st) e^(–t) sin t dt = ∫₀∞ e^(–(s+1)t) sin t dt.
Step 3: Recognize the standard Laplace formula: ∫₀∞ e^(–pt) sin t dt = 1/(p^2 + 1) with p = s + 1.
Step 4: Thus, L{e^(–t) sin t} = 1/[(s+1)^2 + 1].
Final Answer:

Laplace Transform (Definition and Simple Example)

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Common Mistakes

  • Using an incorrect guess for the particular solution in the undetermined coefficients method when the forcing term duplicates a term in the homogeneous solution.
  • Forgetting to multiply the guess by x when the forcing term is a solution of the homogeneous equation.
  • Miscalculating derivatives or sign errors when applying reduction of order.
  • Incorrectly computing the Wronskian or neglecting its role in verifying linear independence.
  • Not checking the compatibility of boundary conditions when solving eigenvalue problems, leading to spurious solutions.