In problems solve the given system of differential equations

In problems solve the given system of differential equations by systematic elimination. dx/dt = 2x - y dx/dt = 4x + 7y dy/dt = x dy/dt = x - 2y dx/dt = -y + t dx/dt - 4y = 1 dy/dt = x - t dy/dt + x = 2 (D^2 + 5)x - 2y = 0 -2x + (D^2 + 2)y = 0 (D+ 1)x + (D - 1)y = 2 3x + (D + 2)y = -1

Solution

given (D+1)x+(D-1)y =2 -------------------------------------------(1)

3x+(D+2)y = -1 -------------------------------------------------------(2) where D = d/df     (f can be x and y)

from (2) x = (-1 -(D+2)y)/3

i.e   x = --(1+y\'+2y)/3 ---------------------------------(3)

x\' = -(y\"+2y\')/3 -----------------------------------------(4)

putting (3) & (4) in (1)\'s simplified form x\'+x+y\'-y =2 gives

-(y\"+2y\')/3-(1+y\'+2y)/3+y\'-y =2

-y\"/3 + y\'(-2/3-1/3 +1) -y(2/3+1) =2

-y\"/3+y\'(0) -y(5/3) =2

y\"/3+5y/3 =-2

y\"+5y =-6

y\"+5y+6 =0