Math 1B: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/~theojf/09Summer1B/ay00+ by0+ cy = g(t)Recall that if y1and y2are solutions to the inhomogeneous equation ay00+ by0+ cy = g(t),then y1− y2is a solution to the homogeneous equation ay00+ by0+ cy = 0. Therefore, the generalsolution to ay00+ by0+ cy = g(t) is:y = yp+ C1y1+ C2y2where y1and y2are linearly independent solutions to the homogeneous equation, and ypis anyparticular solution to the inhomogeneous equation. Thus, we can solve inhomogeneous lineardifferential 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 generalizeswell to higher-order differential equations, but it does not work for all functions g(t). The ideais 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 etsin t, weget et(sin t + cos t), et(2 cos t), et(2 cos t − 2 sin t), etc., all of which are linear combinations of thefunctions etsin t and etcos t. So by “cycle through a finite list” I mean that a function f(t) isspecial 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 correspondingfinite set of functions. Then af00(t) + bf0(t) + cf(t) is definitely a linear combination of the samefinite list of functions. Conversely, let’s say that g(t) is a special function and {g1, . . . , gn} its finitelist. Then for any constants A1, . . . , An, the function f(t) = A1g1(t) + · · · + Angn(t) is special, andit’s reasonable to hope that we find constants A1, . . . , Anso that af00(t)+bf0(t)+cf(t) = g(t), sincefinding 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 outthat the only way we could fail to find a solution in the above paragraph is if there are someconstants B1, . . . , Bnso that B1g1(t) + · · · + Bngn(t) is a solution to the homogeneous equationay00+by0+c = 0. In this case, we have to expand our set {g1, . . . , gn}. We can use the fact, though,that (tg)0= g +tg0, and expand the set by multiplying each member by t. For a second-order lineardifferential 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 differentialequations:• If g(t) = ektp(t), where p is a polynomial of degree n, guess yp(t) = ektq(t), where q is adegree-n polynomial with undetermined coefficients.• If g(t) = ektp(t) cos mt or ektp(t) cos mt, guess yp(t) = ektq(t) cos mt + ektr(t) sin mt.• If g(t) = g1(t) + g2(t), it might be simpler to solve each equation ay00+ by0+ cy = g1(t) anday00+ by0+ 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 equationay00+ by0+ cy = 0, you may have to multiply those terms by t or t2.11. § Write a trial solution for the following differential equations. Do not determine the coeffi-cients.(a) y00+ 9y0= 1 + xe9x(b) y00+ 3y0− 4y = (x3+ x)ex(c) y00+ 2y0+ 10y = x2e−xcos 3x (d) y00+ 4y = e3x+ x sin 2x2. § Find the general solution for the following differential equations:(a) y00− 4y0+ 5y = e−x(b) y00+ 2y0+ y = xe−x3. § Solve the initial value problem:y00+ y0− 2y = x + sin 2x, y (0) = 1, y0(0) = 04. Use the method of undetermined coefficients to find the general solution to the followingfirst-order linear differential equation:y0= eaxsin bxwhere a, b are constants. How would you normally solve this differential equation? Whichmethod do you prefer?5. Use the method of undetermined coefficients to find the general solution to the followingfirst-order linear differential equation:y0= xneaxwhere a is a constant and n is a positive integer. How would you normally solve this differentialequation? Which method do you prefer?6. A spring with spring constant k, mass m, and damping constant c is hung vertically, so thatit experiences a constant downward force mg, where g is the acceleration due to gravity. Findthe 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 withsprings, 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 Ω, an inductor with L = 2 H, and a capacitorwith C = 0.0025 F. Let’s assume that the initial charge on the capacitor is 0, and that theinitial current is 0.(a) § If we add a battery to the circuit that applies a constant potential of 12 V, how willthe circuit respond?(b) How will the circuit respond if we instead use a power source that applies an alternatingpotential 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 thatall derivatives of f + g are linear combinations of members of the list.(b) Prove that the product fg is a special function by finding a finite list of functions sothat all derivatives of fg are linear combinations of members of the list. Hint: thinkabout the product rule.Exercises marked with an § are from Single Variable Calculus: Early Transcendentals for UC Berkeley by James
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