9.2E: Higher Order Constant Coefficient Homogeneous Equations (Exercises)

Q9.2.1

In Exercises 9.2.1-9.2.14 find the general solution.

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Q9.2.2

In Exercises 9.2.15-9.2.27 solve the initial value problem. Graph the solution for Exercises 9.2.17-9.2.19 and 9.2.27.

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Q9.2.3

28. Find a fundamental set of solutions of the given equation, and verify that it is a fundamental set by evaluating its Wronskian at .

Q9.2.4

In Exercises 9.2.29-9.2.38 find a fundamental set of solutions.

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Q9.2.5

39 It can be shown that

where the left side is the Vandermonde determinant and the right side is the product of all factors of the form with and between and and .>

1. Verify (A) for and .
2. Find the Wronskian of .

40. A theorem from algebra says that if and are polynomials with no common factors then there are polynomials and such that This implies that for every function with enough derivatives for the left side to be defined.

1. Use this to show that if and have no common factors and then .
2. Suppose and are polynomials with no common factors. Let , …, be linearly independent solutions of and let , …, be linearly independent solutions of . Use (a) to show that is a linearly independent set.
3. Suppose the characteristic polynomial of the constant coefficient equation has the factorization where each is of the form and no two of the polynomials , , …, have a common factor. Show that we can find a fundamental set of solutions of (A) by finding a fundamental set of solutions of each of the equations and taking to be the set of all functions in these separate fundamental sets.

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1. Show that if where and are polynomials of degree , then where and are polynomials of degree .
2. Apply (a) times to show that if is of the form (A) where and are polynomial of degree , then
3. Use Equation 9.2.17 to show that if then
4. Conclude from (b) and (c) that if and are arbitrary polynomials of degree then is a solution of
5. Conclude from (d) that the functions are all solutions of (C).
6. Complete the proof of Theorem 9.2.2 by showing that the functions in (D) are linearly independent.

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1. Use the trigonometric identities to show that
2. Apply (a) repeatedly to show that if is a positive integer then
3. Infer from (b) that if is a positive integer then
4. Show that (A) also holds if or a negative integer. HINT: Verify by direct calculation that Then replace by in .
5. Now suppose is a positive integer. Infer from (A) that if and then (Why don’t we also consider other integer values for ?)
6. Let be a positive number. Use (e) to show that and

43. Use (e) of Exercise 9.2.42 to find a fundamental set of solutions of the given equation.

44. An equation of the form where , , …, are constants, is an Euler or equidimensional equation. Show that if then

In general, it can be shown that if is any integer then

where , …, are integers. Use these results to show that the substitution (B) transforms (A) into a constant coefficient equation for as a function of .

45. Use Exercise 9.2.44 to show that a function satisfies the equation on if and only if the function satisfies Assuming that , , , are real and , find the possible forms for the general solution of (A).