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Practice Exercises on Differential Equations, Exercises of Differential Equations

Harvard UniversityDifferential Equations

Practice exercises on differential equations. The exercises cover topics such as Fourier series, linear operators, inner products, and the heat equation. The answers to the exercises are provided at the end of the document. useful for students studying differential equations and preparing for exams.

Typology: Exercises

2022/2023

Uploaded on 05/11/2023

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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,… (ak cos(kx) + bk 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 bk cos(kx) - k aksin(kx)). Show 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
1
!
!"
"
#
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,
pf3
pf4
pf5
pf8

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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) = a 0 + ∑k=1,2,… (ak cos(kx) + bk 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 bk cos(kx) - k aksin(kx)). Show 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!^1 !" "

# 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.

  1. 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?

  1. A ubiquitous operator in quantum mechanics sends a function, f, on (-∞, ∞) to the function T(f) = – f´´ + x^2 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.
  2. 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, x^2 } in terms of a and b.
  3. 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 utt – c^2 uxx = 0. b) At what time t ≥ 0 does the function x → u(t, x) have its maximum at x = 10?
  4. 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.
  5. Let u(t, x) denote a solution to the heat equation ut = uxx 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.

  1. Let f and g denote any two continuous functions on [-π, π] and let 〈f, g〉 denote their inner product, 〈f, g〉 =!^1 !" "

# f(x)g(x) dx.

a) Let ε > 0 be any given number. Use the inequality |ab| ≤ 12 (a^2 + b^2 ) on the integrand for suitable a and b to prove that |〈f, g〉| ≤ 12 ε 〈f, f〉 + 12 ε−^1 〈g, g〉. b) Use the preceding inequality with ε = 〈g, g〉1/2^ 〈f, f〉−1/2^ to prove that |〈f, g〉| is never greater than 〈f, f〉1/2^ 〈g, g〉1/2.

  1. Let T denote the operator that sends a smooth function h on (-∞, ∞) to the function T(h) = h´´´´ - 5h´´ + 4h.
  1. Find a solution to the heat equation ut = uxx for t ≥ 0 and x ∈ [-π, π] that is zero at both x = π and x = - π and equals sin(x)(1 + cos(x)) at t = 0.
  2. Find a function, u(t, x), that obeys ut = 4uxx for t ≥ 0 and x ∈ [-π, π] that obeys the initial conditions u(0, x) = sin(2x) + sin(3x) - sin(4x) and vanishes at x = - π and x = π at all t ≥ 0.
  3. Find the form for the general solution of the equation f´´´ + 2f´´ - 3f´ = 0. Then write down a function that obeys this equation with f(0) = 1, f´(0) = 2 and f´´(0) = 3.
  4. Find a function u(x, y) that obeys uxx + uyy = 0 where 0 ≤ x ≤ π and 0 ≤ y ≤ π plus the boundary conditions u(x, 0) = u(x, π) = sin(x) + 2sin(2x) and u(0, y) = u(π, y) = 0.
  5. Find a solution to the heat equation ut = 4uxx on the interval [-π, π] that obeys the initial condition u(0, x) = 2 sin(2x) - 3sin(3x) + 4sin(4x) and obeys for all t ≥ 0 the boundary conditions u(t, π) = u(t, - π) = 0.
  6. Give a basis for the vector space of function that obey f´´´ - 3f´´ = 0 and f(0) = f(1).
  7. Give two distinct solutions to the equation f´´´ - 2f´´ + 2f´ = 0 that obey both f(0) = 0 and limx→∞ f(x) = 1.
  8. Written below are four infinite sequence of the form {a 1 , a 2 , …, an, …} of real numbers. Three of these sequences have the property that the a 1 sin(x) + a 2 sin(2x) + ··· + an sin(nx) + ··· is the Fourier series of a continuous function on the interval [-π, π]. Meanwhile, there is one sequence that does not have this property. Find the latter sequence. a) {1, 12 , 14 , …, ( 12 )n, … }. b) {1, 1, (^32) !, (^) 4!^2 , … , ( (^) n!^2 ), … } c) {1, - 1, 1, - 1, … , (-1)n, … } d) {1, - 12 , 14 , … , (- 12 )n, … }
  9. Write down continuous functions f(x) and g(x) that are defined where – π ≤ x ≤ π and have all of the following properties: a) f(x) = g(x) where x ≤ 0. b) f(0) = g(0) = 1. c) The inner product of f and g,!^1 !" "

# f(x)g(x) dx, is zero.

  1. Give the Fourier series of the function sin( 12 x) for – π ≤ x ≤ π.
  2. Define a linear map from the space of functions on [-π, π] to R by sending any given function f to!^1 !" "

# f(x) sin(2x) dx.

a) Prove that the range of this map is the whole of R. b) Exhibit an orthonormal basis for the kernel of this map. Answers:

1.! 4 - ∑k=1,3,… (^) !k^2 2 cos(kx) - ∑k=1,2,… (-1)k^1 ksin(kx).

  1. (^)!^2 - (^)!^4 ∑k=2,4,… (^) k (^21)! 1 cos(kx).
  2. The coefficient in front of sin(kx) is!^1 !" "

# 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.

  1. The coefficient in front of sin(kx) is!^1 !" "

# f´(x) sin(kx) dx and and so an integration

by parts finds this equal to - (^)!^1 k !" "

# f(x) cos(kx) dx which is – kak. On the otherhand,

the coefficient for cos(kx) is!^1 !" "

# f´(x) cos(kx) dx and an integration by parts finds

that this is (^)!^1 k !" "

# f(x) sin(kx) dx +

1 (^) !(-1) k (^) (f(π) – ƒ(-π)). This equals kb k given that f(π) = f(-π). The integral for the constant term is (^)! 1 " 2 (f(π) – f(-π)) which is zero.

  1. A basis is {e−4x, ex} and the desired element is f(x) = (^) e 1 !+ e^ e! 4 e−4x^ -^1 +e^ ! 4 (^) e! e! 4 e x (^).
  2. A basis is {cos(2x), sin(2x)} and the desired element is f(x) = cos(2x) - 12 sin(2x).
  3. Take the function 1 and the set {sin(x), sin(2x), …, }.
  4. This is the identity operator, it acts to send f → f for any f.
  5. a) T(f) = - (f´ + xf)´ + x(f´ + xf). b) All such functions are multiples of e ! x^2 / 2 .
  6. The orthogonal projection in this case is – a x – 2b x^2.
  1. u(x, y) = sinh(2π)-^1 sinh(x+π) sin(x) + 3 sinh(6π)-^1 sinh(3x + 3π) sin(3y)
  2. u(x, y) = cosh(π)-^1 cosh(x) sin(x)
    • [sinh(6π)-^1 sinh(3x- 3 π) + sinh(3π)-^1 sinh(3x)] sin(3x)
  3. These all have the form cos(kx)(a eky^ + b e-ky) where a and b are constants.
  4. f 1 = 1, f 2 = x and f 3 = 1-x.
  5. 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.
  6. Differentiating u finds that h must obey h + x^2 hyy at all y. Plug in x = 0 to see that h(y) = 0 for all y.
  7. Since sin(x)(1 + cos(x)) = sin(x) + 12 sin(2x), take u(t, x) = e-t^ sin(x) + 12 e-4t^ sin(2x).
  8. u(t, x) = cos(2t) sin(2x) + 14 sin(4t) sin(4x) + 13 sin(3t) cos(3x).
  9. The general form is f(x) = a + b ex^ + c e-3x. Of these, f(x) = - 43 + 94 ex^ + (^) 1 2^1 e-3x^ obeys the stated conditions.
  10. u(x, y) = ( 1! e! " e " ! e (^)! "e y (^) + e ! (^) " 1 e ! " e (^) " !e
  • y) sin(x) + 2 (^1!^ e ! 2 " e 2 " ! e (^)! 2 "e 2y (^) + e 2! (^) " 1 e 2! " e (^) " 2 !e
  • 2y) sin(2x).
  1. u(t, x) = e-16t^ 2sin(2x) – e-12t^ 3sin(3x) + e64t^ 4sin(4x).
  2. A basis is {1, e3x^ + (1 – e^3 )x}.
  3. Any function of the form 1 – e-x^ cos(x) + c e-x^ sin(x) for c ∈ R has the desired properties. Choose any two distinct values for the constant c.
  4. If {a 1 , a 2 , … } are the coefficients from the sine terms in a Fourier series of some continuous function, then the sum of their squares, a 12 + a 22 + ···, must be finite since it is equal to the integral of the square of the function. This means, in particular, that the sequence must converge to zero. Such is not the case for the sequence in c).
  5. There are infinitely many such pairs. Here is one: Take f(x) = 1. Meanwhile, take g(x) to equal 1 where x ≤ 0 and to equal 1- (^)!^4 x where x ≥ 1.
  1. sin( 12 x) = - 1 6! ∑k=1,2,… (-1)k^4 kk (^2)! 1 sin(kx)
  2. a) If r ∈ R, then the map sends the function r sin(2x) maps to r. b) {1, cos(x), cos(2x), cos(3x), …., sin(x), sin(3x), sin(4x), ….}.