Homework SourseConsider the second order dierential equation y3y 2y 10 cos
Consider the second order dierential equation
y\'\'-3y\'+ 2y = 10 cos(2x) (1)
(a) Find the general solution of the homogeneous equation corresponding
to (1).
(b) Find a particular solution of the inhomogeneous equation (1).
(c) Solve the initial value problem given by (1) and initial conditions
y(0) = 0, y\'(0) = 0.
please solve step by step
y\'\'-3y\'+ 2y = 10 cos(2x) (1)
(a) Find the general solution of the homogeneous equation corresponding
to (1).
(b) Find a particular solution of the inhomogeneous equation (1).
(c) Solve the initial value problem given by (1) and initial conditions
y(0) = 0, y\'(0) = 0.
please solve step by step
Solution
a.>for complementary function:
(D^2-3D+2)y=0
roots are d=2,1
yc.f = Ae2x+ Bex
b.>for particular integral
let yP =Mcos2x +Nsin2x
putting yp in the actual equation and comparing the coefficients of sin2x and cos 2x we get
N=3M and -2M-6N=10
which gives N=-3/2 and M=-1/2
yp = -(1/2 )cos2x -(3/2)sin2x
c.> general solution =Ae2x+Bex -(1/2 )cos2x -(3/2)sin2x +C
