Question

solue
the erran diferential
equation .
d2y/dx2_
_3dy/dx+2y=4ex​

Answer 1

Explanation:

d^2y/dx^2) -3(dy/dx)+2y=e^-3x

put=d/dx=D

(D^2–3D+2)y=0

D^2–3D+2=0

D^2–2D-D+2=0

D(D-2)-1(D-2)=0

(D-1)(D-2)=0

D=1,2

Thus the complementary function for this differential equation is

y=c1e^x+c2e^2x where c1,c2 are constant terms

now for particullar integral ,

P.I.=e^-3x/D^2–3D+2

put D=a (i.e. put the constant power term to which e is raised to )

here a=-3

thus putting D=-3 in the equation ,

e^-3x/(-3)^2–3(-3)+2

=e^-3x/20

thus the solution is: complementary function+P.I

y=c1e^x+c2e^2x+e^-3x/20