Homework Set 2 (Due Aug. ****3rd, ****2024)
The references used to prepare this problem set are “Elementary Differential Equations with Boundary Value Problems” (William Trench) and “Introduction to Differential Equations” (Michael Taylor). In the following problems, y' = dt/dy, y'' = dt2/d2y, and y''' = dt3/d3y.
● What does it mean for a pair of functions {f1 (t), f2 (t)} to be linearly independent? How do we generalize this definition to a set of more than two functions, say {f1 (t), f2 (t), f3 (t)}?
o Show that for constants a and b such that b ≠ 0, the functions eat cos(bt) and eat sin(bt) are linearly independent.
o Explain why the functions t|t| and t2 are linearly independent on ( − ∞, ∞). Can we specify a certain interval for t on which t|t| and t2 are linearly dependent?
■ Hint: Graph these two functions.
● Consider the constant coefficient third order differential equation
y''' + a2y'' + a1y' + a0y = 0, where a2, a1, a0 are constants. Suppose the initial conditions are y'' (t0) = C2, y' (t0) = C1, y(t0) = C0. Find an equivalent first order system and then infer that there are at most three linearly independent solutions.
o Hint: Notice the initial conditions and draw an analogy to the second order case.
● Regarding the previous problem with equation y''' + a2y'' + a1y' + a0y = 0 having initial
conditions y'' (t0) = C2, y' (t0) = C1, y(t0) = C0, show that the eigenvalues for the matrix in the first-order system are obtained by solving λ3 + a2λ2 + a1λ + a0λ = 0.
● Consider the equation t2y'' − 6ty' + 12y = 0 with initial conditions y' (t0) = C1 and y(t0) = C0.
o Hypothesizing that there are solutions of the form y(t) = tr, where r is a constant to be determined, find two solutions and show that they are linearly independent (see the first bullet point).
o Suppose t0 ≠ 0. Construct infinitely many solutions when the interval f代写 Elementary Differential Equations or t includes zero. If the interval is of the form (a, b) where either a > 0 or b < 0, explain why we obtain a unique solution.
o In contrast to the constant coefficient case, say y'' − 2y' − 3y = 0 with initial
conditions y' (t0) = β and y(t0) = α like in a lecture example, what might suggest non-uniqueness of solutions in this problem when the interval includes zero?
■ Hint: Write the given second order equation as a first order system to bring it to a
form which looks like “dt/dy = f(t, y)” and then think about the “rule-of-thumb” we mentioned regarding ∂y/∂f.
● Use the method of variation ofparameters to find a solution to each of the following:
。
。
。 y''' + y = et with initial conditions y'' (t0) = C, y' (t0) = b, and y(t0) = a
。 y'' + y = tan t with initial conditions y' (1) = 2 and y(1) = 4
● Draw the phase portraits for each of the following (show all your work):
。 y'' + 2y' + y = 0
。 y'' + 7y' + 6y = 0
。 y'' − 2y' + 2y = 0
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