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If f(D)y=0 be a linear differential equations with constant coefficient then it's auxiliary equation is?
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If f(D)y=0 be a linear differential equations with constant coefficien...
Auxiliary Equation of Linear Differential Equations with Constant Coefficients


Linear differential equations with constant coefficients are of the form:

f(D)y = any(n) + an-1y(n-1) + ... + a1y' + a0y = 0

where an, an-1, ..., a1, a0 are constants and D is the differential operator.


Finding the Auxiliary Equation


The auxiliary equation is obtained by assuming a solution of the form:

y = ert

Substituting this into the differential equation gives:

f(D)y = anrnert + an-1rn-1ert + ... + a1rert + a0ert = 0

Dividing by ert gives:

anrn + an-1rn-1 + ... + a1r + a0 = 0

This is known as the auxiliary equation.


Example


Consider the differential equation:

y'' - 3y' + 2y = 0

The auxiliary equation is:

r2 - 3r + 2 = 0

This can be factored as:

(r-1)(r-2) = 0

Therefore, the roots are:

r1 = 1 and r2 = 2

The general solution is then:

y = c1et + c2e2t

where c1 and c2 are constants.
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If f(D)y=0 be a linear differential equations with constant coefficient then it's auxiliary equation is?
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