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y’’+ay’+by=0 be differential equation where a & b are constants, show that the general solution tends to zero as x tends to infinity iff both a & b are positive

y’’+ay’+by=0 be differential equation where a & b are constants, show that the general solution tends to zero as x tends to infinity iff both a & b are positive

Grade:12th pass

1 Answers

Vikas TU
14149 Points
7 years ago
This is a higher level education which is taght in coleges/universities.
Let y = e^mx be the soln. of the given eqn.
then
we get quadratic,
m^2 + am + b = 0
m1 = (-a + root(a^2 – 4b))/2
m2 =  (-a - root(a^2 – 4b))/2
As roots are distinct,
The general soln. is:
 
Yg = c1e^(m1x) + c2 e^(m2x)
where c1 and c2 are the constants.
 
or
Yg =  c1e^x*(-a + root(a^2 – 4b))/2   + c2e^x*(-a - root(a^2 – 4b))/2
as x tends to infinity
Yg tends to zero hence.
for a>0
and
b>0 

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