Consider the System: dx/dt = x + 2y and dy/dt = 2x + y. Check if the given pairs
ID: 2945221 • Letter: C
Question
Consider the System: dx/dt = x + 2y and dy/dt = 2x + y.
Check if the given pairs of functions are solutions.
A) (x,y) = (e^(-t) + e^(-3t), -e^(-t) + e^(-3t))
B) (x,y) = (-e^(-t) + e^(3t), e^(-t) + e^(3t))
C) Find the second-order differential equation satisfied by x(t). Choose one of a-e
a. x'' - 2x' + 3x = 0
b. x'' - 4x' - 2x = 0
c. x'' - 2x' - 3x = 0
d. x'' + 4x' - 2x = 0
e. None of these
D) Find the second-order differential equation satisfied by y(t). Choose one of a-e
a. y'' - y' + y = 0
b. y'' + y' + y = 0
c. y'' - y' - y = 0
d. y'' + y' - y = 0
e. None of these
Explanation / Answer
x' = x + 2y
y' = 2x + y
A) (x,y) = (exp(-t) + exp(-3t), -exp(-t) + exp(-3t))
x' = x + 2y = -exp(-t) + 3exp(-3t)
y' = 2x + y = exp(-t) + 3exp(-3t)
(x',y') = (-exp(-t) - 3exp(-3t), exp(-t) -3exp(-3t))
Not a solution
B) (x,y) = (-exp(-t) + exp(3t), exp(-t) + exp(3t))
x' = x + 2y = exp(-t) + 3exp(3t)
y' = 2x + y = -exp(-t) + 3exp(3t)
(x',y') = (exp(-t) + 3exp(3t), -exp(-t) + 3exp(3t))
It's a solution
C) x' = x + 2y
y = (1/2)(x' - x)
y' = (1/2)(x" - x')
y' = 2x + y
(1/2)(x" - x') = 2x + (1/2)(x' - x)
x" - x' = 4x + x' - x
x" - 2x" - 3x = 0
Choice (c) is correct
D) y' = 2x + y
x = (1/2)(y' - y)
x' = (1/2)(y" - y')
x' = x + 2y
(1/2)(y" - y') = (1/2)(y' - y) + 2y
y" - y' = y' - y + 4y
y" - 2y' - 3y = 0
Choice (e) is correct