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crimeas [40]
3 years ago
6

Find a homogeneous linear differential equation with constant coefficients whose general solution is given by

Mathematics
1 answer:
frez [133]3 years ago
6 0

Answer:

y" + 2y' + 2y = 0

Step-by-step explanation:

Given

y=c_1e^{-x}cosx+c_2e^{-x}sinx

Required

Determine a homogeneous linear differential equation

Rewrite the expression as:

y=c_1e^{\alpha x}cos(\beta x)+c_2e^{\alpha x}sin(\beta x)

Where

\alpha = -1 and \beta = 1

For a homogeneous linear differential equation, the repeated value m is given as:

m = \alpha \± \beta i

Substitute values for \alpha and \beta

m = -1 \± 1*i

m = -1 \± i

Add 1 to both sides

m +1= 1 -1 \± i

m +1= \± i

Square both sides

(m +1)^2= (\± i)^2

m^2 + m + m + 1 = i^2

m^2 + 2m + 1 = i^2

In complex numbers:

i^2 = -1

So, the expression becomes:

m^2 + 2m + 1 = -1

Add 1 to both sides

m^2 + 2m + 1 +1= -1+1

m^2 + 2m + 2= 0

This corresponds to the homogeneous linear differential equation

y" + 2y' + 2y = 0

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