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- Consider the subspace W of D, given by W = span(sin x, cos x). (a) Show that the differential operator D maps W into itself. (b) Find the matrix of D with respect to B = {sin x, cos x}. (c) Compute the derivative of f(x) = 3 sin x - 5 cos x indirectly and verify that it agrees with f'(x) as computed directly.Let C2(-∞,∞)={f(x) in C(-∞,∞)|f''(x) exists for all x} be the set of differential functions. Show this is a subspace of C(-∞,∞).Show that the solution set S of the second-order differential equation y" + ay' + by = 0 is a subspace of F
- Obtain the orthogonal trajectories of xy = C and compute for the positive value of y at the point (3,2) when x=4. Use 2 decimal places.Determine two functions, defined on the interval ( − ∞ , ∞ ) , whose Wronskian is given by W ( f 1 , f 2 ) = e 2 x . Are the functions that you found linearly independent on ( − ∞ , ∞ ) ? How do you know?Suppose at (0,1) the function f(x,y) has linearization L = 2x + 3y -3. Let v = <1,0>. What are f(0,1) and Dvf(0,1)?
- Find the linearization of f(x)=1−∫3x+124/(3+t)dt at x=2. upper limit of integral = x+1 lower limit of integral = 3Determine two functions, defined on the interval (−∞,∞)(−∞,∞), whose Wronskian is given by W(f1,f2)=e2xW(f1,f2)=e2x. Are the functions that you found linearly independent on (−∞,∞)(−∞,∞)? How do you know?(a) Let vt) be a differentiable vector valued function of t. If v. (dy/dt) = 0 for all t, can we say anything about |v|?Justify your answer and give it a meaningful interpretation.
- How is the linearization of f (x, y) at (a, b) defined?Let C[−π, π] be the vector space of functions that are continuous over the interval [−π, π]. Find the dimensionof the subspace of C[−π, π] that is spanned by the set {1, cos(2x), cos^2(x)}.Consider the subspace W of D, given by W = span(e2x, e2x cos x, e2x sin x). (a) Find the matrix of D with respect to B = {e2x, e2x cos x, e2x sin x}. (b) Compute the derivative of f(x) = 3e2x - e2xcosx+ 2e2x sin x indirectly, using and verify that it agrees with f' (x) as computed directly.