Exercise 7.2.16 Use the dimension theorem to prove Theorem 1.3.1: If A is an m x...
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
Use the properties of a Cauchy-Euler system to find a general solution of the given system. 2 9 tx'(t) X(t), t> 0 -2 13 For t>0, any Cauchy-Euler system of the form tx' = Ax with A an nxn constant matrix has nontrivial solutions of the form x(t) = t’u if and only if ris an eigenvalue of A and u is a corresponding eigenvector. x(t) =
Use the properties of a Cauchy-Euler system to find a general solution of the given system. 3 7 tx'(t)= X(t), t> - 3 13 For t>0, any Cauchy-Euler system of the form tx' = Ax with A an nxn constant matrix has nontrivial solutions of the form x(t)= t'u if and only ifr is an eigenvalue of A and u is a corresponding eigenvector. x(t) = 0
Use induction and Pascal's identity to prove that (7) = 2" where n > 0.
Use the properties of a Cauchy-Euler system to find a general solution of the given system. 8 5 tx' (t) = X(t), t> 0 - 8 21 For t>0, any Cauchy-Euler system of the form tx' = Ax with A an nxn constant matrix has nontrivial solutions of the form x(t) = t'u if and only if ris an eigenvalue of A and u is a corresponding eigenvector. X(t) = 0
Use the properties of a Cauchy-Euler system to find a general solution of the given system. 8 5 tx' (t) = X(t), t> 0 - 8 21 For t>0, any Cauchy-Euler system of the form tx' = Ax with A an nxn constant matrix has nontrivial solutions of the form x(t) = t'u if and only if ris an eigenvalue of A and u is a corresponding eigenvector. X(t) = 0
Now assume that f(0) = 0 and f'(0) = 0. Prove that if f is twice differentiable and If"(x) < 1 for all x E R then 22 Vx > 0, f(x) < 2
Use induction to prove that 0–0 4j3 = n4 + 2n3 + n2 where n > 0.
(7) Green's Theorem for Work in the Plane F(x, y) =< M, N >=< x, y2 > C: CCW once about y = vw and y = x W = | <M,N><dx,dy>= | Mdx + Ndy CZ CZ (70) Parametrize the path Cy: along the curve y = vw from (1,1) to (0,0) in terms of t. (70) Use this parametrization to find the work done. (7e) Confirm Green's Theorem for Work. (7) Green's Theorem for Work in the Plane...
Let F = < x-eyz, xexx, z?exy >. Use Stokes' Theorem to evaluate slice curlĒ ds, where S is the hemisphere x2 + y2 + z2 = 1, 2 > 0, oriented upwards.