In this problem you will solve the nonhomogeneous system y?' = [[4 2] [-9 -2]] y? + [-2 e^t e^t] A. Write a fundamental matrix for the associated homogeneous system ? = [[e^t(-cos(3t)+sin(3t)) e^t(-cos(3t)-sin(3t))] [3 e^t cos(3t) 3 e^t sin(3t)]] B. Compute the inverse ?^{-1} = [[e^{-t} sin(3t) 0.471 e^{-t} sin(3t+pi/4)] [-e^{-t} cos(3t) -0.471 e^{-t} cos(3t+pi/4)]] C. Multiply by g? and integrate ? ?^{-1} g? dt = [ (first component) + c1 (second component) + c2 ] D. Give the solution to the system y? = [ (first fundamental column) ] c1 + [ (second fundamental column) ] c2 + [ (particular solution) ]
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471 \\ 2 & -0.471 \end{pmatrix} \begin{pmatrix} \sin(t) + 0.471\sin(3t + \frac{\pi}{4}) \\ 2\cos(3t) - 0.471\cos(3t + \frac{\pi}{4}) \end{pmatrix} \] Show more…
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C. Multiply by g→ and integrate D. Give the solution to the system
Vincenzo Z.
In this problem you will solve the nonhomogeneous system y' = [[-2, 5], [-9, 4]] y + [[-2e^t], [-e^t]] A. Write a fundamental matrix for the associated homogeneous system B. Compute the inverse C. Multiply by g and integrate (Do not include c1 and c2 in your answers). D. Give the solution to the system (Do not include c1 and c2 in your answers). If you don't get this in 2 tries, you can get a hint.
Sri K.
In this problem you will use undetermined coefficients to solve the nonhomogeneous equation y'' - 5y' + 6y = 9e^{2t} - 6te^{2t} - (6t + 13) with initial values y(0) = 1 and y'(0) = 7. A. Write the characteristic equation for the associated homogeneous equation. (Use r for your variable.) r^2-5r+6=0 B. Write the fundamental solutions for the associated homogeneous equation. y1 = e^(2t) y2 = e^(3t) C. Write the form of the particular solution and its derivatives. (Use A, B, C, etc. for undetermined coefficients. Y = (A*t^2+B*t)e^(2t)+Ct+D Y' = (2At^2+2Bt+2At+B)e^(2t)+C Y'' = (4At^2+4Bt+8At+4B+2A)e^(2t) D. Write the general solution. (Use c1 and c2 for c1 and c2). y = c1*e^(2t)+c2*e^(3t)+(3t^2-3t)e^(2t)-(t+18) E. Plug in the initial values and solve for c1 and c2 to find the solution to the initial value problem. y = 46e^(2t)-27e^(3t)+(3*t^2-3t)e^(2t)-(t+18) Hint: No fractions are required in the solution or answer to this problem.
Madhur L.
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