In this problem, you will solve the non-homogeneous differential equation
y ′′+12y′+32y= sin(e4x).
(1) Let C1 and C2 be arbitrary constants. The general solution to the related homogenous differential equation y′′+12y′+32y=0 is the function yh(x)=C1y1(x)+C2y2(x)=C1 _____ +C2 _____.
Note: the order in which you enter the answers is important; that is,
C1f(x)+C2g(x)≠C1g(x)+C2f(x).
(2) The particular solution yp(x) to the differential equation y′′+12y′+32y=sin(e4x)
is of the form yp(x)=y1(x)u1(x)+y2(x)u2(x) where u′1(x)= _____ and u′2(x)= _____.
(3) It follows that u1(x)= _____ and u2(x)= _____: thus yp(x)= _____.
(4) The most general solution to the non-homogeneous differential equation y′′+12′+32y= sin (e4x) is y=C1 _____ + C2 _____ + _____.
Variation of parameters
Differential equation of form ϕ(D)y=f(x)where,D=ddx has solution y=yc+yp
Where, complimentary function yc is depend on auxiliary equation and
particular integral is depend on function f(x).
In this method y p is depend on yc
and represented as yp=u1y1+u2y2
with u′1=−y2fn(x)wandu′2=y1f(x)wandw= (y1y2y′1y′2)=y1y′2−y2y′