Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b) k k k=. Write the following in closed form. (c) k= ( ) k kx k. k! (a) x! + x4 4! x6 x3 +... (b) 4 + + 3x5 6! 6! 0! + 4x7 4! +... (c) + x + x4 4! + 4x + x8 8! + 6x3 3 + x! +... 3. Write the following as a Maclaurin series in simplified form. Maclaurin Series: f (x) = f (0) + f (0) + f (0)! x + f (0) 3! x 3 + f (4) (0) x 4 +... 4! (a) y = sin x (b) y = e x (c) y = x 3 4x Introduction to Power Series If the power series a n (x c) n has radius of convergence R > 0, then the k= coefficients a n are simply the Taylor coefficients of f at c: a n = f (n) (c) n!. Solve the following first and second order linear differential equations: (write your series solutions in closed form) (a) Solve using the methods learned earlier this semester. (b) Solve using the Taylor Series Method. (c) Solve using the Power Series Method. (d) Determine the radius of convergence of the resulting series. (e) Check that the results are equivalent. i. y + 4y = 0, y(0) = c 0 ii. y + 3y = 0, y(0) = c 0 iii. (x )y + y = 0, y(0) = c 0 iv. y + 4y = 0, y(0) = 0, y (0) = 3 v. y + y = x, y(0) = c 0, y (0) = c vi. y y + y = 0, y(0) = 0, y (0) = Series Solutions Near Ordinary Points. Find the general solution for the following differential equations and determine a lower bound for the radius of convergence of the power series solution around c = 0. (a) y + xy + y = 0 (b) (x + 3)y 7xy + 6y = 0 (c) y + x y + xy = 0 Translated Series Solution. Solve the initial value problems by first making the appropriate substitution t = x a. Also state the radius of convergence for your solution. (a) (x x )y 6(x )y 4y = 0 y() = 0 y () = (b) (x + 6x)y + (3x + 9)y 3y = 0 y( 3) = 0 y ( 3) = Series Solutions Near Regular Singular Points. Verify that x = 0 is a regular singular point for the following differential equations. (a) x ( + x)y + x(4 x )y + ( + 3x)y = 0 (b) x 4 y + (x sin x)y + ( cos x)y = 0. Bessel s equation of order zero: x y + xy + x y = 0 has a regular singular point at x = 0.
Differential Equations - Homework For Test 3 (a) Use the power series method (and let c 0 = ) to show that one ( ) solution of this differential equation is J 0 (x) = n x n n (n!) which is referred to as the Bessel function of order zero of the first kind. (b) Now use Mathematica to solve the differential equation. Note how the solution is written. Use the series command to get the first five or six terms. These should match your solution if you were to expand your power series solution. (c) Note that Mathematica also gives another solution Y 0 (x) which is referred to as the Bessel function of order zero of the second kind. Use Mathematica to Plot both Bessel functions and identify which plot goes with each solution. 3. Bessel s equation of order one: x y + xy + (x )y = 0 has a regular singular point at x = 0. (a) Use the power series method (and let c = /) to show that one solution of this differential equation is J (x) = x ( ) n x n which is referred to as the Bessel n!(n + )!n function of order one of the first kind. (b) Now use Mathematica to solve the differential equation. Note how the solution is written. Use the series command to get the first five or six terms. These should match your solution if you were to expand your power series solution. (c) Note that Mathematica also gives another solution Y (x) which is referred to as the Bessel function of order one of the second kind. Use Mathematica to Plot both Bessel functions and identify which plot goes with each solution. 4. What pattern differences do you see in your graphs comparing Bessel functions of the first kind to Bessel functions of the second kind? 5. A special case of a non-constant coefficient problem is of the form ax y + bxy + cy = 0 This equation is referred to as the Cauchy-Euler Equidimensional Equation. It is the simplest case of a second order equation with regular singular points. In general, a differential equation with a regular singular point and real roots to its indicial equation will have at least one Frobenius series solution which is of the form: c n x n+r (a) Use the substitution v = ln x to convert the equation into one that has constant coefficients. (b) Find the characteristic equation of the differential equation. Assume the three possible cases: distinct roots, equal roots, and complex roots and write the general solution for each case. Make sure your solution is with respect to the independent variable x; not v. (c) Now use the substitution c n x n+r and show that we end up (when n = 0) with the characteristic equation ar + (b a)r + c = 0. In this context, it is actually referred to as the indicial equation of the differential equation and the solutions r, r are referred to as the exponents of the differential equation. This method is more general and works for differential equations of the form: x y + xp(x)y + q(x)y = 0. 6. Use the method of Frobenius (substitute solution to the Bessel equation of order /: x y + xy + (x 4 )y = 0 Fourier Analysis Fourier Series c n x n+r ) to find the general. Find the Fourier Series for the following and graph the original function and the first five or six terms of its Fourier Series on the same graph in Mathematica. (a) Find the Fourier Series for a periodic function with period π that is defined from π to π. 0 π < t 0 i. f (t) = 0 < t π ii. f (t) = t iii. f (t) = π < t π π + t π t 0 0 0 t π
Differential Equations - Homework For Test 3 3 (b) Find the Fourier Series for a periodic function. 3 < t < 0 i. f (t) = 0 < t < 3 ii. f (t) = t < t < (c) Find the Fourier Cosine Series and the Fourier Sine Series for the following functions. i. f (t) = t 0 < t < ii. f (t) = t 0 < t < π iii. f (t) = t(π t) 0 < t < π Complex Fourier Series. Find the following for the function f (x) = e x x (a) The complete Fourier Series (b) The complex Fourier Series (c) Convert the complex Fourier coefficients into real Fourier coefficients. Your answer should match part a. Applications of Fourier Series Boundary Value Problems. Solve the following boundary value problems: (a) y + 4y = 0, y(0) =, y( π 4 ) = 0 (b) y + 4y = 0; y(0) =, y(π) = (c) y + 4y = 0; y(0) =, y(π) = 3 (d) y + 5y = 0; y (0) = 6, y (π) = 9 (e) y + 4y = 0; y(0) = 0, y(π) = 0. Find all eigenvalues and eigenfunctions for the following boundary value problems: (a) y + λy = 0, y(0) = 0, y(π) = 0 (b) y + λy = 0, y (0) = 0, y (π) = 0 Partial Differential Equations. Show that the following functions are solutions to the partial differential equation: x f x + y f y = f (a) f (x, y) = 3x 0. y 0.8 (b) f (x, y) = 8 xy (c) f (x, y) = 3x. Consider the heat diffusion equation: T t = κ T where T(y, t) is a y function which describes the temperature T (in F) at position y (in meters) at time t (in hours). What are the units of the constant κ, usually called the thermal diffusivity? 3. At time t = 0 the temperature of a concrete slab is described by the function T(x) = 0 cos π x + 3x + 40 where x = 0 at one end of the 0 slab and x = 0 at the other end. The way the temperature will change over time is controlled by the heat diffusion equation: T t = κ T x. At what rate is the temperature changing at x = 0 and t = 0 if κ = meters squared per second? 4. For what value of c does the function T(y, t) = 3e ct sin πy satisfy the heat diffusion equation T t = κ T where κ = 4? y 5. The function T(y, t) = e 4t sin 5y is a solution of the heat diffusion equation T t = κ T y where κ is in feet per second. What are the units of the constants, 4, and 5? 6. The temperature of a concrete slab must satisfy the heat diffusion equation T t = κ T where κ = meters squared per hour, with the x boundary conditions that the temperature is always 60 F at the origin and at x = 0 meters. Which of the following could be T(x, t)? (a) T = 7e tπ /5 sin π 5 x + 60 (b) T = 8e tπ 8/5 sin π 5 x + 60
Differential Equations - Homework For Test 3 4 (c) T = 60 (d) All of the above 7. Use the method of separation of variables with the constant of separation λ to solve the following 5 u t = u, 0 < x < 0, t > 0, u(0, t) = u(0, t) = 0, u(x, 0) = 4x x 8. A copper rod (κ =.5) 50 cm long with insulated lateral surface has initial temperature u(x, 0) = x, and the two ends of the rod are insulated. (a) Find u(x, t) (b) Use u(x, t) to approximate the temperature at x = 0 after minute. (c) Approximate how long it will take for the temperature at x = 0 to become 45 C. Laplace Transforms Introduction to Laplace Transforms Definition of Laplace Transform. Find the Laplace transform of the following by using the definition. (a) f (t) = for t 0 (b) Derive the Laplace transform of y(t) = sinh kt (c) Find the Laplace transform of: 5 if 0 t 0 f (t) = if 0 < t 5 0 otherwise (f) The function represented by the graph below: y Table of Laplace Transforms 3 4 5. Use the table of Laplace transforms to find the Laplace transform of the following. You can use Mathematica to verify your answer: LaplaceTransform[f(t),t, s] (a) 3e t + sin 3t (b) t + 3t (c) ( + x) 3 (d) x 3 5(x ) x (e) x sin 3x (f) cosh 4t (g) t sin(t π/6) (h) t cos t (i) First write the function as an infinite sum of unit step functions and then find the Laplace transform of the sum using a table. Then use the geometric series formula to show that L f (t)} = s( + e s ) f (t) 3 4 5 6 t (d) f (t) = t a where a is a real number and a > (e) The unit step function: u a (t) = u(t a) = 0 if t < a if t a
Differential Equations - Homework For Test 3 5 (j) Express the square wave function g(t) in terms of the function f (t) from the previous problem. You can then easily find the Laplace of g(t) using the answer from the previous problem. Show L g(t)} = e s s( + e s ) = s tanh s g(t) h(t) 3 4 5 6 t Laplace Transform Using Theorem: L e at f (t) } = F(s a) 0-3 4 5 6 t. Find the Laplace transform of the following: (a) f (t) = t 4 e πt (b) f (t) = e t sin 3πt -. Use the table of Laplace transforms to find the inverse Laplace transform of the following. You can use Mathematica to verify your answer: InverseLaplaceTransform[f(s),s,t] (a) s + 3 (b) s + 3 Laplace Transform Using Theorem: (c) 3s + 4 s + 4 (d) s /. Find the Laplace transform of the following: (a) f (t) = te at (b) f (t) = t sin kt (c) f (t) = t cos kt (d) Derive the Laplace transform of f (t). L g (t) } = sg(s) g(0) (e) Take the derivative of the triangular wave function graphed below. The answer should be a recognizable function that you have taken the Laplace of already. Use the theorem to find the Laplace of the given function h(t).. Find the Inverse Laplace transform of the following: 3s + 5 (a) F(s) = s 6s + 5 s + (b) F(s) = s + 4s + 5 3. Use Partial Fractions to find the Inverse Laplace transform of the following: s (a) F(s) = + s 3 s 8s 5 s (b) F(s) = s + 7s + 0 Inverse Laplace Transform Using Convolution Theorem: L f (t) g(t)} = F(s)G(s). Find the convolution of the two functions (a) f (t) = t g(t) = e at (b) f (t) = g(t) = e at. Find the Inverse Laplace Transform (a) F(s) = (s )(s + 4)
Differential Equations - Homework For Test 3 6 (b) F(s) = s(s 3) (c) X(s) = W(s)F(s) Piecewise Continuous Functions. Find the Laplace transform of the following.. Sketch the graphs of the following. (a) t u(t ) (a) (b) (t ) u(t ) (c) t u(t ) t u(t ) (d) (t ) u(t ) (t ) u(t ) (e) u(t ) + u(t ) u(t 3) (b) - 3 4 5 6 (f) u(t ) u(t ) + 3 u(t 3). Write the following in terms of unit step functions, u(t a). 3 4 5 6 (a) y - (c) (b) y 3 4 5 x - 3 4 5 6 3 4 5 6 x (d)...... 3 4 5 Laplace Transforms Using Theorem: L u(t a) f (t a)} = e as F(s) -. Find the Laplace transform of the following. (a) (t ) u(t ) (b) t u(t ) (c) (sin t) u(t π ) (d) (sin t) u(t π 6 ) 3. Find the Laplace transform of f (t) = u(t a + ɛ) u(t a ɛ), where a > ɛ, using the definition of the Laplace transform.
Differential Equations - Homework For Test 3 7 Laplace Transforms Using Theorem:. Find the Laplace transform of the following: (a) f (t) = t sin kt (b) f (t) = te t cos 3t Applications of Laplace Transforms L t n f (t)} = ( ) n F (n) (s) Solve the following differential equations using Laplace Transforms. x x 6x = 0, x(0) =, x (0) =. x + 4x = sin 3t, x(0) = x (0) = 0 3. x x x = 0, x(0) = 0, x (0) = 4. x + x = cos 3t, x(0) =, x (0) = 0 5. x + 6x + 34x = 0, x(0) = 3, x (0) = 6. y + 4y + 4y = t, y(0) = y (0) = 0 7. x + 6x + 34x = 30 sin t, x(0) = x (0) = 0 8. x + 4x + 8x = e t, x(0) = x (0) = 0 9. tx + x + tx = 0, x(0) =, x (0) = 0 0. tx (4t + )x + (t + )x = 0, x(0) = 0. y + 3y + y = 0, y(0) =, y (0) = 0. y + y + 5y = 0, y(0) =, y (0) = 3. y y = u(t ) u(t 4), y(0) =, y (0) = 0, y (0) =, y (0) = 0 4. y + ω y = cos t, ω = 4, y(0) =, y (0) = 0 5. tx + (t )x x = 0, x(0) = 0 6. x t if 0 t 6 + 5x = f (t), x(0) = 0, where f (t) = 0 otherwise 7. x + 4x = f (t), x(0) = x cos t if 0 t π (0) = 0, f (t) = 0 otherwise 8. x + 4x + 5x = δ (t ), x(0) = 0, x (0) = 0 9. x + 4x = 8δ π (t), x(0) = 3, x (0) = 0 0. x + 4x + 4x = + δ(t ), x(0) = x (0) = 0. x + 4x + 5x = δ(t π) + δ π (t), x(0) = 0, x (0) =. Derive the solution: x(t) = t equation: x + x + x = f (t) x(0) = x (0) = 0 0 τe τ f (t τ) dτ for the differential 3. Consider a mass-spring system with m =, k =, no damping and forcing function f (t) = 5 k= δ(t (k )π). The initial displacement is zero and the initial velocity is zero. Solve the initial value problem and sketch the solution on the interval 0 t 50π. Explain the relationship between the graph and f (t). Theory Problems. True or False. The Laplace transform method is the only way to solve some types of differential equations.. Which of the following differential equations would be impossible to solve using the Laplace transform? (a) dc dr = c + 3 sin(r) + 8 cos (3r + ) (b) d f dx 00 d f dx = 8 ln x (c) g (b) = g (d) p (q) = 4 q + 96p p 3. Assume we have a homogeneous linear differential equation with constant coefficients and initial conditions y(0) = y (0) = 0. True or False. We add the nonhomogeneous driving term f (t) to the equation and have y(t) as a solution. If instead we add the nonhomogeneous driving term f (t), then the solution will be the function y(t).
Differential Equations - Homework For Test 3 8 4. 5. 0 5 0 5 (δ 3 (t) + δ 6 (t) 3δ 8 (t) + 5δ (t)) dt = (δ 6 (t) δ 7 (t) δ 8 (t) δ 9 (t)) dt = 6. We have a system modeled as an undamped harmonic oscillator, that begins at equilibrium and at rest, so y(0) = y (0) = 0, and that receives an impulse force at t = 4, so that it is modeled with the equation y = 9y + δ 4 (t). Find y(t) without using pencil/paper. Using the theory you learned this semester you should be able to calculate the solution in your head with minimal calculations. 7. We have a system modeled as an undamped harmonic oscillator, that begins at equilibrium and at rest, so y(0) = y (0) = 0, and that receives an impulse force at t = 5, so that it is modeled with the equation y = 4y + δ 5 (t). Find y(t) without using pencil/paper. Using the theory you learned this semester you should be able to calculate the solution in your head with minimal calculations. 8. For many differential equations, if we know how the equation responds to a nonhomgeneous forcing function that is a Dirac delta function, we can use this response function to predict how the differential equation will respond to any other forcing function and any initial conditions. For which of the following differential equations would this be impossible? values of p and q. What will be the solution of d y dt + p dy + qy = 0, dt with y(0) = 0 and y (0) = 5?. Let ζ(t) be the solution to the initial-value problem d y dt + p dy dt + qy = δ 0(t), with y(0) = y (0) = 0. Now suppose we want to solve this problem for some other forcing function f (t), so that we have d y dt + p dy + qy = f (t). Which of the following is a correct dt expression for L y}? (a) L y} = 0 (b) L y} = L f } L ζ} (c) L y} = L f } L ζ} (d) L y} = L f ζ}. Let ζ(t) = e 3t be the solution to the initial-value problem d y dt + p dy dt + qy = δ 0(t), with y(0) = y (0) = 0 for some specific values of p and q. Now suppose we want to solve this problem for the forcing function f (t) = e t, so that we have d y dt + p dy + qy = f (t). dt Find y(t). (a) p (q) = 4 q + 96p p (b) d m dm 8 00 + 4nm = dn dn ln n (c) d f dx 00 d f dx = 8 ln x (d) dc dr = c + 3 sin(r) + 8 cos (3r + ) 9. Let ζ(t) be the solution to the initial-value problem d y dt + p dy dt + qy = δ 0(t), with y(0) = y (0) = 0. Find the Laplace transform of ζ(t). 0. Let ζ(t) = e 3t be the solution to the initial-value problem d y dt + p dy dt + qy = δ 0(t), with y(0) = y (0) = 0 for some specific