Differential Equations Homework Just Case 2 In the case that

Differential Equations Homework

(Just Case 2)

In the case that one or more of the terms in the derivatives of f(x) du V part of the general solution of the associated homogeneous equation, what may attempt yp = x^s middot y\'p, where y\'p is what one would try if there were no duplication and the smallest integer such that none of the terms of yp is a solution of the \'associated homogeneous equation.

Solution

y\"-5y\'+6y=e^4x

p^2-5p+6 is factorised as , (p-2)(p-3)
or
(d/dx-2)(d/dx-3)y=e^4x
gives
(d/dx-2)y=(d/dx-3)^-1 e^4x

(d/dx-a)^-1(f(x))
is given by e^ax*integral{f(x)*e^-ax}
using this formula
(d/dx-2)y=e^3x *integral(e^4x*e^-3x)
(d/dx-2)y=e^3xintegral(e^x)
(d/dx-2)y=e^3x*(e^x+c)=e^4x+ce^3x where c is a constant
y=(d/dx-2)^-1{e^4x+ce^3x}

y=e^2x integral {e^-2x(e^4x+ce^3x}
y=e^2x integral {e^2x+ce^x}
y=e^2x{(e^2x)/2+ce^x+d}

y=(e^4x)/2+ce^3x+de^2x where c and d are constants