Math 1B Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Summer1B ay 00 by 0 cy g t Recall that if y1 and y2 are solutions to the inhomogeneous equation ay 00 by 0 cy g t then y1 y2 is a solution to the homogeneous equation ay 00 by 0 cy 0 Therefore the general solution to ay 00 by 0 cy g t is y yp C1 y1 C2 y2 where y1 and y2 are linearly independent solutions to the homogeneous equation and yp is any particular solution to the inhomogeneous equation Thus we can solve inhomogeneous linear differential equations provided we have some methods to guessing particular solutions to them The method we will outline is called the method of undetermined coefficients It generalizes well to higher order differential equations but it does not work for all functions g t The idea is as follows Certain special functions namely polynomials exponentials sines and cosines and sums and product of these have the special property that upon repeated differentiation the functions cycle through a finite list For example if we repeatedly differentiate et sin t we get et sin t cos t et 2 cos t et 2 cos t 2 sin t etc all of which are linear combinations of the functions et sin t and et cos t So by cycle through a finite list I mean that a function f t is special in this sense if there is a finite set of functions f1 t fn t so that all derivatives of f t including f itself are linear combinations of elements of the set In particular let s say that f t is a special function and f1 t fn t the corresponding finite set of functions Then af 00 t bf 0 t cf t is definitely a linear combination of the same finite list of functions Conversely let s say that g t is a special function and g1 gn its finite list Then for any constants A1 An the function f t A1 g1 t An gn t is special and it s reasonable to hope that we find constants A1 An so that af 00 t bf 0 t cf t g t since finding such constants requires only that we solve a system of n linear equations in n unknowns Of course not every system of n linear equations in n unknowns has a solution It turns out that the only way we could fail to find a solution in the above paragraph is if there are some constants B1 Bn so that B1 g1 t Bn gn t is a solution to the homogeneous equation ay 00 by 0 c 0 In this case we have to expand our set g1 gn We can use the fact though that tg 0 g tg 0 and expand the set by multiplying each member by t For a second order linear differential equation you may have to do this at most twice for an nth order differential equation n times All in all we get the following rules for guessing the form of particular solutions to differential equations If g t ekt p t where p is a polynomial of degree n guess yp t ekt q t where q is a degree n polynomial with undetermined coefficients If g t ekt p t cos mt or ekt p t cos mt guess yp t ekt q t cos mt ekt r t sin mt If g t g1 t g2 t it might be simpler to solve each equation ay 00 by 0 cy g1 t and ay 00 by 0 cy g2 t separately and add the answers If any y t of the form of the guess yp t is itself a solution to the complementary equation ay 00 by 0 cy 0 you may have to multiply those terms by t or t2 1 1 Write a trial solution for the following differential equations Do not determine the coefficients a y 00 9y 0 1 xe9x b y 00 3y 0 4y x3 x ex c y 00 2y 0 10y x2 e x cos 3x d y 00 4y e3x x sin 2x 2 Find the general solution for the following differential equations a y 00 4y 0 5y e x b y 00 2y 0 y xe x 3 Solve the initial value problem y 00 y 0 2y x sin 2x y 0 1 y 0 0 0 4 Use the method of undetermined coefficients to find the general solution to the following first order linear differential equation y 0 eax sin bx where a b are constants How would you normally solve this differential equation Which method do you prefer 5 Use the method of undetermined coefficients to find the general solution to the following first order linear differential equation y 0 xn eax where a is a constant and n is a positive integer How would you normally solve this differential equation Which method do you prefer 6 A spring with spring constant k mass m and damping constant c is hung vertically so that it experiences a constant downward force mg where g is the acceleration due to gravity Find the equilibrium solution i e find the position at which the spring will hang without moving Then find the general solution Explain how why for the purposes of solving problems with springs we can ignore gravity if we measure the displacement from the equilibrium solution rather than from the location in which the spring doesn t apply any force 7 A series circuit contains a resistor with R 40 an inductor with L 2 H and a capacitor with C 0 0025 F Let s assume that the initial charge on the capacitor is 0 and that the initial current is 0 a If we add a battery to the circuit that applies a constant potential of 12 V how will the circuit respond b How will the circuit respond if we instead use a power source that applies an alternating potential of 12 sin 50t s V 8 Let f and g be special functions so that all the derivatives of f are linear combinations of f1 fm and all derivatives of g are linear combinations of g1 gn a Prove that the sum f g is a special function by finding a finite list of functions so that all derivatives of f g are linear combinations of members of the list b Prove that the product f g is a special function by finding a finite list of functions so that all derivatives of f g are linear combinations of members of the list Hint think about the product rule Exercises marked with an are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart 2
View Full Document
Unlocking...