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# 96ä¸Šå¾®åˆ†æ–¹ç¨‹ æœŸä¸­

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2x3 -e 6' delve the given initial-value problem by using an appropriate substitution j2/ Solve the given initial-value problem "'*Q% xdx + (x2 + 4 y)dy = 0, y(4) = 0 3. The number 0 is a critical point of the autonomous differential equation -4 4L stable? Semi-stable? Unstable? - -6 (7%) 9 7 '7 \ 5' %fL / 7- /Find a on:-parameter 'family of solutions for the differential equation 2 dy 2 2 x -+xy+4=x y - _ 2 dx , where Y1 - is a known solution of the & x \ L \$ d3~(x) + 7 d2~(x) +16 d~(x) + 12y(x) = 0 5 (G dx dx dx ,y (-- (a) Find a fundamental set of solutions for the differential equation. You need to _ /G (V f 78- ,)\$ &ow that the set of sol~~tions you find is a fundamental set of solutions. (8%) 43 cC 7 2+*9 (b) Find the general solution ~(x) ofthe differential equation. (2%) c / -v -IF h3f7d =-w-F~ (~-t2)(1-+~)tr-+3) +(dWt-[r m'+w%@.~ 3qL+1 2.m f ( L

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LrL' -' - L*~-I 0 (7 - / f le-r 'L dFind the general solution of 2e2y T ,p L d2y(x) dy(x) 2 x 2 -2 dC2 dx = 3e + 4x , with the initial condition: 7. Consider the system of differential equations: dx(t) dy(t) 2- + - - y(t) = sin t dt dt - - dy (1) "(') + 6x(t) + - + 5 = cost dt dt L Find the general solution, ~(t) &
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96ä¸Šå¾®åˆ†æ–¹ç¨‹ æœŸä¸­

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