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Find the general solution of the differential equation d^2y/dx^2 - 5*dy/dx + 4y = 2x

Solve complimentary function: Let y = e^mxthen,
d²y/dx² - 5dy/dx + 4y = 0
m²e^mx - 5me^mx + 4e^mx = 0 (substituting for y)
e^mx(m² - 5m + 4) = 0
e^mx(m - 4)(m - 1) = 0
Therefore m=4 and m=1, so the c.f. is y = Ae^4x+ Be^x where A,B are constants

Solve particular integral: Let y = ax + b and substitute into the differential equation
0 - 5a + 4(ax + b) = 2x
4ax + (4b - 5a) = 2x
Therefore 4a=2 and 4b-5a=0 so a=1/2, b =5/8

Hence the general solution is y = c.f + p.i = Ae^4x+ Be^x + 1/2 x + 5/8

Answered by Peter M. • Further Mathematics tutor

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Find the general solution of the differential equation d^2y/dx^2 - 5*dy/dx + 4y = 2x

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