Question

Find the particular integral of the equation (D2 + 2D + 1)y=e^ -xlogx by using the method of variation of parameter. Solve the differential equationne (my) = 20​

Answer 1

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For the equation y’’ -2y’ + y =(e^x)lnx the characteristic equation is

(m -1 )^2 =0 , roots m = 1 , 1.The solution of the homogeneous part

is yh = C1e^x + C2xe^x . The functions y1(x) = e^x , y2(x) = xe^x are independent

integrals with WronsKian W(x) = e^2x. The variation of parameters method

gives the particular integral by

yp = -y1(x)[ Integral of ( y2(x)e^x(lnx)dx/W(x) )] +

+ y2(x) [ Integral of ( y1(x)e^x(lnx)dx/W(x) ) ]. From

Integral of lnxdx = xlnx - x and

Integral of xlnxdx = (1/2)x^2lnx - x^2/4 one obtains

yp = e^x[(1/2)lnx -(3/4)x^2 ].