Practice Exercises on Differential Equations

What follows are some exerices to help with your studying for the part of the final exam on differential equations. In this regard, keep in mind that the exercises below are not necessarily examples of those that you will see on the final exam. Even so, if you understand how to do these, you should do fine on the differential equation portion of the final. The answers are provided at the end.

Exercises:

1. Find the Fourier series of the function on [-π, π] that equals x where x ≥ 0 and zero where x < 0.

2. Find the Fourier series of the function on [-π, π] given by x ® |sin(x)|.

3. Let f(x) denote a function on [-π, π] with the property that f(x) = f(-x) for all x. Explain why there are no sine functions in the Fourier series of f.

4. By its very definition, the Fourier series of a smooth function x ® f(x) on [-π, π]

has the form f(x) = a₀ + ∑_k=1,2,… (a_k cos(kx) + b_k sin(kx)). When computing the Fourier series of the derivative, f´(x), there is the inevitable temptation to exchange orders of differentiation and summation and so conclude that f´(x) has the Fourier series ∑_k=1,2,… (k b_k cos(kx) - k a_ksin(kx)). Prove that this is the correct answer when f(π) = f(-π) by computing the relevant integrals.

5. Find a basis for the kernel of the linear operator f ® f´´ + 3f´ – 4f on the space of smooth functions on [0, 1]. Find an element in the kernel of this operator that obeys f(0) = 1 and f(1) = -1.

6. Find a basis for the kernel of the linear operator f ® f´´ + 4f on the space of smooth functions on [-π, π]. Find an element in the kernel of this operator that obeys f(0) = 1 and f´(0) = -1.

7. An inner product on the space of continuous functions on [-π, π] is defined as in the

Differential Equation Handout using the rule that has the inner product between

functions f and g equal to f(x)g(x) dx. Exhibit a non-zero function, g, and an infinite, orthonormal set such that g is orthogonal to each element in this set.

8. Let A denote the linear operator f ® f´ and let B denote the linear operator f ® x f,

where f here is any smooth function on (-∞, ∞). What is the operator AB – BA?

9. A ubiquitous operator in quantum mechanics sends a function, f, on (-∞, ∞) to the

function T(f) = –f´´ + x² f – f.

a) Suppose that f is a function that obeys f´+ x f = 0. Prove that T(f) = 0.

b) Write down all functions f that obey f´ + x f = 0.

10. Reintroduce the inner product from Problem 7 on the space of functions in [-π, π].

Let f be any function on [-π, π] that vanishes at the endpoints. Write the orthogonal projection of f onto the span of {1, x} as a + bx. Give the orthogonal projection of f´(x) onto the span of {1, x, x²} in terms of a and b.

11. Let x ® f(x) be a smooth function such that f(1) = 2 while f(x) < 2 if x ≠ 1. Let c > 0

be a constant.

a) Prove that the function u(t, x) = f(x-ct) obeys the version of the wave equation given by u_tt – c² u_xx = 0.

b) At what time t ≥ 0 does the function x ® u(t, x) have its maximum at x = 10?

12. Let T denote the operator that sends a smooth function, h, on (-∞, ∞) to the function

T(h) = h´´ + 2h´ + h. Exhibit two different functions in the kernel of T that both equal 1 at x = 0.

13. Let u(t, x) denote a solution to the heat equation u_t = u_xx on [-π, π] whose x-derivative

is zero at both x = π and x = -π for all t ≥ 0.

a) Explain why the function t ® u(t,x) dx is constant.

b) Explain why the function t ® u(t, x)² dx is non-increasing as a function of t.

In your explanations to both a) and b), you don’t have to justify exchanging orders of

differentiation and integration.

14. Let f and g denote any two continuous functions on [-π, π] and let áf, gñ denote their

inner product, áf, gñ = f(x)g(x) dx.

a) Let e > 0 be any given number. Use the inequality |ab| ≤ (a² + b²) on the integrand for suitable a and b to prove that |áf, gñ| ≤ e áf, fñ + e^-1 ág, gñ.

b) Use the preceding inequality with e = ág, gñ^1/2 áf, fñ^-1/2 to prove that |áf, gñ| is never greater than áf, fñ^1/2 ág, gñ^1/2.

15. Let T denote the operator that sends a smooth function h on (-∞, ∞) to the function

T(h) = h´´´´ - 5h´´ + 4h.

a) Exhibit a basis for the kernel of T.

b) Give the dimension of the subspace of functions in kernel(T) that are zero at x = 1.

16. Let T denote the operator that sends a smooth function h on [0, ∞) to the function

T(h) = h´ + h. Suppose that g is a smooth function on (-∞, ∞). Denote by h the

function x ® e^-x .

a) Prove that T(h) = g.

b) What is the range of T?

c) What is the dimension of the kernel of T?

17. Let n be any positive integer and let {f₁, …, f_n} be any finite set of continuous

functions on [-π, π]. Prove that there is a non-zero, continuous function g on this same interval that is orthogonal to each f_k.

18. Exhibit an infinite dimensional subspace in the space of continuous functions on the

interval [-π, π] whose complement as defined by the inner product in Problems 7 and 14 is also infinite dimensional.

19. Write down a solution, u(t, x), to the equation u_tt – 4u_xx = 0 for t ≥ 0 and x Î [-π, π]

that obeys u(0, x) = sin(x) cos(x) and u_t(0, x) = 0.

20. a) Give a solution, u(t, x), to the equation u_tt – 4u_xx = 0 for t ≥ 0 and x Î [-π, π]

that obeys u(0, x) = 0 and u_t(0, x) = 1.

b) Give a solution to this same equation that obeys u(0, x) = 0 and u_t(0, x) = sin²(x).

21. Let k denote a positive integer. Find all solutions to the Laplace equation u_xx + u_yy = 0

that have the form u(x, y) = cos(kx) h(y).

22. Exhibit three functions on (-∞, ∞) with the following properties: The first is a linear

combination of the second and third, the first is never zero, the second is zero only at the origin, and the third is zero only at x = 1.

23. Suppose that f is never zero on [-π, π]. Prove that every function on [-π, π]

that is orthogonal to f with respect to the inner product in Exercise 14 must be zero at some point in x Î (-π, π).

24. Explain why there is no function of the form u(x, y) = x² h(y) that solves the Laplace

equation u_xx + u_yy = 0 except in the case that h is identically zero.

25. Find a solution to the heat equation u_t = u_xx for t ≥ 0 and x Î [-π, π] that is zero at

both x = π and x = -π and equals sin(x)(1 + cos(x)) at t = 0.

Answers:

1. - ∑_k=1,3,… cos(kx) - ∑_k=1,2,… (-1)^k sin(kx).

2. -∑_k=2,4,… cos(kx).

3. The coefficient in front of sin(kx) is f(x) sin(kx) dx and this is zero since the

contribution from the x ≥ 0 part of the integral is minus that from the x ≤ 0 part.

4. The coefficient in front of sin(kx) is f´(x) sin(kx) dx and and so an integration by

parts finds this equal to -kf(x) cos(kx) dx which is –ka_k. On the otherhand, the coefficient for cos(kx) is f´(x) cos(kx) dx and an integration by parts finds that this is kf(x) sin(kx) dx + (-1)^k (f(π) – ƒ(-π)). This equals kb_k if f(π) = f(-π). The integral for the constant term is (f(π) – f(-π)) which is zero.

5. A basis is {e^-4x, e^x} and the desired element is f(x) = e^-4x - e^x .

6. A basis is {cos(2x), sin(2x)} and the desired element is f(x) = cos(2x) - sin(2x).

7. Take the function 1 and the set {sin(x), sin(2x), …, }.

8. This is the identity operator, it acts to send f ® f for any f.

9. a) T(f) = - (f´ + xf)´ + x(f´ + xf) .

b) All such functions are multiples of e .

10. The orthogonal projection in this case is –a x – 2b x².

11. a) Use the Chain Rule to deduce that u_t = -cf´|_x-ct and u_tt = c² f´´|_x-ct. Meanwhile,

u_x = f´|_x-ct and u_xx = f´´|_x-ct.

b) This occurs at t = 9/c.

12. One is e^-x and another is (1+x) e^-x.

13. a) The time derivative of the indicated integral is u_t(t,x) dx. Since the heat equation is obeyed, this is u_xx dx. Integrating by parts finds the latter equal to

u_x(t, π) – u_x(t, -π) which is assumed to be zero.

b) The time derivative of the indicated integral in this case is 2u u_t dx. Use the heat equation to write this as 2u u_xx dx. Then integrate by parts to equate the latter integral with -2u_x² dx + 2 (u u_x)|_(t,π) – 2 (u u_x)|_(t,-π). Since this last part is

zero and the first part is non-positive, the function of t here can not be increasing.

14. a) Write a = e^1/2 f(x) and b = e^-1/2 g(x) to see that |f(x)g(x)| ≤ e f²(x) + e^-1 g²(x).

Do this for each value of x to see that f(x)g(x) dx has absolute value that is no greater thane f²(x) dx + e^-1g²(x) dx. This is the desired

conclusion.

b) Taking this value for e makes e áf, fñ = e^-1ág, gñ = áf, fñ^1/2ág, gñ^1/2.

15. a) The corresponding polynomial is r⁴ – 5r² + 4 = (r² –1)(r² – 4). Thus, the roots are

r = ±1 and r = ±2. This implies that kernel(T) = Span{e^-x, e^x, e^-2x, e^2x}.

b) This is a 3-dimensional vector subspace spanned by {e^-x-e^-2e^x, e^-2x-e^-3e^x, e^2x-ee^x}.

16. a) Differentiating finds h´ = -e^-x + e^-x(e^xg(s))|_s=x = -h + g.

b) Let g be any smooth function. Since h = e^-x obeys T(h) = g, so g is in

the range of T. Thus, all smooth functions are in T’s range.

c) The kernel of T is one dimensional, spanned by e^-x.

17. The orthogonal projection from the span of {sin(x), sin(2x), …, sin((n+1)x)} to the

span of {f₁,…, f_n} is a linear map from an (n+1) dimensional space to an n dimensional one, so it must have a positive dimensional kernel. Any function in this kernel is orthogonal to all f_k.

18. {sin(x), sin(2x), …} is orthogonal to {, cos(x), cos(2x), …}.

19. u(t, x) = cos(4t) sin(2x).

20. a) u(t, x) = t.

b) u(t, x) = (t - sin(4t) cos(2x)).

21. These all have the form cos(kx)(a e^ky + b e^-ky) where a and b are constants.

22. f₁ = 1, f₂ = x and f₃ = 1-x.

23. The function f is either positive or negative but never zero. If g is orthogonal to f, then

the product fg must change signs along [-π, π] and so g must be zero somewhere between –π and π.

24. Differentiating u finds that h must obey h + x²h_yy at all y. Plug in x = 0 to see that

h(y) = 0 for all y.

25. Since sin(x)(1 + cos(x)) = sin(x) + sin(2x), take u(t, x) = e^-t sin(x) + e^-4t sin(2x).