solve the differential equation


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  1. Solve the characteristic equation to get the homogeneous solution as
    yh(x)=C1*e^x+C2*e^3x, or
    y1(x)=e^x, y2(x)=e^(3x)

    Use variation of parameters to find the particular solution, namely, assume the particular solution to be
    which implies:
    v1'(x)y1(x)+v2'(x)y2(x)=0 ...(1)
    v1'(x)y1'(x)+v2'(x)y2'(x)=sin(2x)cos(x) ...(2)

    Substitute y1'(x) and y2'(x) in (2) to get
    and by substituting v2'(x) into (1), we get

    Integrate v1'(x) and v2'(x) to find v1(x) and v2(x).
    Substitute v1(x) and v2(x) into (0) to find yp(x).

    The general solution is:
    =C1*e^x+C2*e^(3x) + (3sin(x)+6cos(x)-sin(3x)+2cos(3x))/60

    Check my arithmetic or typo.

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