代写Math 4b Homework Set 2代做Statistics统计
- 首页 >> C/C++编程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 for 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