The indicated function y1x is a solution of the associated h

The indicated function y1(x) is a solution of the associated homogeneous equation. Use the method of reduction of order to find a second solution y2(x) of the homogeneous equation and a particular solution yp(x) of the given nonhomogeneous equation. y\'\' 3y\' + 2y = 13e^3x; y1 = e^x

Solution

let y2(x) = v(x) y1(x) be another solution to homogenous equation,

=>
y2 = ve^x,

y\'2 = ve^x + v\'e^x= e^x(v+v\')

y\"2 = e^x(v+2v\'+v\")

=>
e^x(v+2v\'+v\") -3e^x(v+v\') + 2e^x(v) = 0

=>
e^x(-v\' + v\") = 0

=>

v\" = v\'

=>
v = e^x

=>
y2(x) = e^x * e^X = e^2x

=>
general solution of homogenous equation is y = c1e^x + c2e^2x

let y = ce^(3x) be a solution of the given equation

=>
y\" = 9ce^3x, y\' = 3ce^3x

=>

(9ce^3x) -3(3ce^3x) + 2(3ce^3x) = 13e^3x

=>

c = 13/2

=>
y = 13(e^3x) /2 is a particular solution

=>
y = c1e^x + c2e^2x + 13e^(3x) /2 is general solution of the equation