Skip to content
Quick Guides

Differential Equations - Second-Order Equations

Preview 10 of 45 pages

See the first 10 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    Study Guide Differential Equations – Second - Order Equations 1. Second - Order Homogeneous Equations 1.1 What Does “Homogeneous” Mean Here? The term homogeneous differential equation is used in two different ways, depending on context. First meaning (earlier topic) A first - order equation of the form is called homogeneous if ( M ) and ( N ) are homogeneous functions of the same degree. Second meaning (used much more often) A differential equation of any order is called homogeneous if all terms involving the unknown function are on one side , and the other side is zero. For example: • y'' - 2y' + y = 0 (homogeneous) • y'' - 2y' + y = x (not homogeneous) This second definition is the one we will focus on in this chapter. 1.2 Homogeneous vs Nonhomogeneous Equations A general second - order nonhomogeneous linear equation looks like If we replace the right - hand side by zero, we get the corresponding homogeneous equation : The equation (**) plays a crucial role in solving (*) .

  2. Page 2
    Read page 2 as text

    Study Guide 1.3 Two Fundamental Theorems These two results explain how solutions of homogeneous and nonhomogeneous equations are related. Theorem A If ( y 1 (x) ) and ( y 2 (x) ) are linearly independent solutions of a linear homogeneous equation, then every solution of that equation can be written as a linear combination of them: This expression is called the general solution of the homogeneous equation. Theorem B If: • ( y p (x) ) is any particular solution of the nonhomogeneous equation, and • ( y h (x) ) is the general solution of the corresponding homogeneous equation , then the general solution of the nonhomogeneous equation is In words: General solution of a nonhomogeneous equation = general solution of the corresponding homogeneous equation • any particular solution The function ( y h ) is also called the complementary function . Example 1: General Solution of a Homogeneous Equation Consider: We are told that

  3. Page 3
    Read page 3 as text

    Study Guide are solutions. Step 1: Form Linear Combinations By Theorem A, every solution has the form: Step 2: (Optional) Verify the Solution Differentiate: Substituting into the equation shows that is satisfied. Final Answer Example 2: Solving a Nonhomogeneous Equation Verify that ( y = 4x - 5 ) satisfies

  4. Page 4
    Read page 4 as text

    Study Guide Step 1: Verify the Particular Solution If ( y = 4x - 5 ) , then: Substitute: So ( y = 4x - 5 ) is a particular solution. Step 2: Solve the Corresponding Homogeneous Equation The homogeneous equation is: We are given two linearly independent solutions: So, Step 3: Apply Theorem B Add the particular solution: Example 3: Trigonometric Solutions First, verify that: solve the homogeneous equation: Both satisfy the equation, and they are linearly independent.

  5. Page 5
    Read page 5 as text

    Study Guide Step 1: Write the Homogeneous Solution By Theorem A: Step 2: Solve the Nonhomogeneous Equation Now consider: By inspection, you can see that ( y = x ) is a particular solution. Step 3: Write the General Solution Using Theorem B: Summary • A homogeneous equation has zero on the right - hand side • Solutions of homogeneous equations combine linearly • Linearly independent solutions form the building blocks of all solutions • For nonhomogeneous equations: o First solve the corresponding homogeneous equation o Then add any particular solution • These ideas apply to linear equations of any order

  6. Page 6
    Read page 6 as text

    Study Guide 2. The Method of Undetermined Coefficients 2.1 Why This Method Exists To solve a nonhomogeneous linear differential equation , we learned (from Theorem B) that: General solution = homogeneous solution + particular solution So the real challenge is often finding a particular solution . When the nonhomogeneous term has a special form, we can use a powerful and systematic technique called the method of undetermined coefficients . 2.2 When Can This Method Be Used? Consider the nonhomogeneous equation The method of undetermined coefficients works only when the nonhomogeneous term ( d(x) ) belongs to a special class of functions called those with a finite family of derivatives . 2.3 What Is a “Finite Family of Derivatives”? A function has a finite family of derivatives if all of its derivatives can be written using only a finite number of functions . Example: A Function With a Finite Family Let Its derivatives cycle: • d' = cos x • d'' = - sin x • d''' = - cos x • d 4 = sin x All derivatives involve only sin x and cos x.

  7. Page 7
    Read page 7 as text

    Study Guide So the family of derivatives is: Example: A Function Without a Finite Family Let Its derivatives keep producing higher and higher powers of sec x and tan x. There is no finite set that contains all derivatives. Conclusion: The method of undetermined coefficients cannot be used when d(x) = tan x 2.4 Functions That Work with This Method The following functions do have finite derivative families and are suitable for this method: 2.5 Linear Combinations A linear combination of functions ( y 1 , y 2 , ……. y n ) is any expression of the form where the ( c i ) are constants.

  8. Page 8
    Read page 8 as text

    Study Guide Key Idea of the Method To find a particular solution, form the most general linear combination of the functions in the family of ( d(x) ) , then substitute it into the differential equation and solve for the constants. Example 1: Identifying a Family If The family is: (The constant 5 is ignored.) Example 2: Product of Functions If • Family of ( x ) : ( {x, 1} ) • Family of (sin 2x): ( sin 2x, cos 2x) So the combined family is: Example 3: Polynomial Forcing Term Find a particular solution of Step 1: Choose a Trial Function The family of ( 5x 2 ) is ( x 2 , x, 1 ) , so try:

  9. Page 9
    Read page 9 as text

    Study Guide Step 2: Substitute and Match Coefficients Substituting into the equation and combining like terms gives: Equating coefficients: Solving: 2.6 Particular Solution Example 4: Trigonometric Forcing Term Solve: Step 1: Choose Trial Function Family of sin x: sin x, cos x Try:

  10. Page 10
    Read page 10 as text

    Study Guide Step 2: Substitute and Solve After substitution: So: This gives: Particular Solution Complete Solution Example 5: Exponential Forcing Term Solve: Try: Substitution gives:

35 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.