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For #1 and #2, find the general solution of the ODE system tX' = AX, t>...
(2) (4 marks.) Find the general solution of the ODE e-2x Y"' + 4y + 4y...
(2) (4 marks.) Find the general solution of the ODE e-2x Y"' + 4y + 4y = X > 1. 2
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5=...
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5= [ 9), P=[7"}] (1) 5= [ • ? ], P=[} >>]
Use the properties of a Cauchy-Euler system to find a general solution of the given system....
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 the properties of a Cauchy-Euler system to find a general solution of the given system....
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....
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
(1 point) Find the general solution, y(t), which solves the problem below, by the method of...
(1 point) Find the general solution, y(t), which solves the problem below, by the method of integrating factors. 6t+y=t", t> 0 dt Put the problem in standard form. Then find the integrating factor, y(t) = and finally find y(t) = (use C as the unkown constant.)
Use the properties of a Cauchy-Euler system to find a general solution of the given system....
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) =
First-Order ODE (a) .Find the general solution of the following ODE: (b). Find the general...
First-Order ODE
(a) .Find the general solution of the following ODE:
(b). Find the general solution (for x > 0) of the ODE :
Hint: try the change of variables u ≜ x, v ≜ y/x.
(c). Find the solution to the ODE
that satisfies y(2) = 15.
Hint: Try separation of variables. For integration,
try partial fraction decomposition.
2Ꮖy 2 Ꭸ , . + <+5 12 , fi - z - ,fix = zu y' = y2...
(1 point) Find the inverse Laplace transform of 2s + 9 $2 + 23 S> 0...
(1 point) Find the inverse Laplace transform of 2s + 9 $2 + 23 S> 0 y(t) =
Consider the signal x(t) = te-atu(t), a > 0 Find to = 1.00 /*(t)?|dt Find to...
Consider the signal x(t) = te-atu(t), a > 0 Find to = 1.00 /*(t)?|dt Find to = 10lx(t)2|dt Can simplify to → %*tx?(t)dt x2(t)dt
(2) (4 marks.) Find the general solution of the ODE e-2x Y"' + 4y + 4y = X > 1. 2
2) Sketch the phase portrait of the system x' (t) = Ax (t) if (a) 5= [ 9), P=[7"}] (1) 5= [ • ? ], P=[} >>]
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 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
(1 point) Find the general solution, y(t), which solves the problem below, by the method of integrating factors. 6t+y=t", t> 0 dt Put the problem in standard form. Then find the integrating factor, y(t) = and finally find y(t) = (use C as the unkown constant.)
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) =
First-Order ODE
(a) .Find the general solution of the following ODE:
(b). Find the general solution (for x > 0) of the ODE :
Hint: try the change of variables u ≜ x, v ≜ y/x.
(c). Find the solution to the ODE
that satisfies y(2) = 15.
Hint: Try separation of variables. For integration,
try partial fraction decomposition.
2Ꮖy 2 Ꭸ , . + <+5 12 , fi - z - ,fix = zu y' = y2...
(1 point) Find the inverse Laplace transform of 2s + 9 $2 + 23 S> 0 y(t) =
Consider the signal x(t) = te-atu(t), a > 0 Find to = 1.00 /*(t)?|dt Find to = 10lx(t)2|dt Can simplify to → %*tx?(t)dt x2(t)dt
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