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Solve: (5+2x)^2 d^2y/dx^2- 6 (5+2x)dy/dx + 8y=6x |
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Answer» For the equation ((5+2x)^2)y’’ - (5+2x)y’ + 8y = 6x , take 5+2x = e^t . Then y’(x) = 2y’(t)e^(-t) , y’’(x) = 4[ y’’(t) - y’(t) ]e^(-2t) and the equation becomes 2y’’ -3y’ + 4y = (3/2)e^t - 15/2.The characteristic polynomium is 2m^2 - 3m + 4=0 roots 3/4 +i((23)^1/2)/4 , 3/4 - i((23)^1/2)/4 . Then the solution of the homogeneous part is yh = (e^3/4t)( C1 cos((23^1/2)/4)t + C2sin((23^1/2)/4)t ). For the particular integral try yp = Ae^t + B . Substituting inthe equation obtain A = 1/2 , B = -15/8. Finally y(t) = (e^3/4t)( C1cos((23^1/2)/4)t + C2sin((23^1/2)/4)t ) +e^t/2 -15/8 . The solution of the equation is obtained with t = ln(5 + 2x). |
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