Equations in $D'(RR)$ -n.2
Solve in $D’(RR) $ the following equations :
a) $(x^2-5x+6)^2 u(x) =0 $
b) $ sin x .u(x) =0 $
c) $x^2(x^2-pi^2).sin x.u(x)=0 $
a) $(x^2-5x+6)^2 u(x) =0 $
b) $ sin x .u(x) =0 $
c) $x^2(x^2-pi^2).sin x.u(x)=0 $
Risposte
Since I didn’t see any solution, I will provide a commented one to exercise b) and just the solutions to a) and c).
First, I recall a theorem very useful .
Let $I$ be an open interval and $psi $ a function $in C^(oo) $ in $I$ , its zeros being all isolated and of finite order.
Then the solutions of the equation : $ psi u = 0 $ in $I $ are given by the distributions of the type :
$ u(x) = sum_(z in ZZ) sum_(i=0)^(w(z)-1) C_(z,i)* delta ^ i (x-z) $, where $ZZ$ is the set of the zeros of $ psi$ , while $w(z) $ is the order of the generic zero $z $ and $C_(z,i) $ are arbitrary constants.
In the case of equation b ) , we have :
$psi = sinx $ , function which is analytic , not constant and therefore all its zeros are isolated , of finite and integer order: the zeros are all and only the points $x_k = kpi $ , with $ k in ZZ$ ; furtherly such $x_k $ are all simple zeros since :
$sin(x_k) = 0 $ , while $ (d sin(x_k))/dx =cos(x_k) ne 0 $ .
The theorem indicates that solutions to equation b) are all distributions $u(x) $ such that :
$u(x) = sum_(k=-oo)^(+oo)C_k delta(x-kpi)$ with $C_k $ being independent constants, since it is : $w(z) = 1,therefore i=0 $.
In addition, we note that for each selection of $C_k $ , the series representing $u(x) $ converges in the sense of distributions.
In a similar way we get the solutions of equations a ), c) :
a) $u(x) = c_1delta(x-2) +c_2 delta’(x-2) +c_3 delta(x-3) +c_4 delta’(x-3)$. $[x=2 ;x=3 $ are roots of order 2 ].
b) $u(x)=a_1 delta’(x)+a_2delta’’(x)+a_3 delta’(x-pi)+a_4delta’(x+pi)+sum_(k in ZZ) c_k(x-kpi)$.$[x=0 $ is a root of order 3 , while $x=+-pi$ are roots of order 2].
First, I recall a theorem very useful .
Let $I$ be an open interval and $psi $ a function $in C^(oo) $ in $I$ , its zeros being all isolated and of finite order.
Then the solutions of the equation : $ psi u = 0 $ in $I $ are given by the distributions of the type :
$ u(x) = sum_(z in ZZ) sum_(i=0)^(w(z)-1) C_(z,i)* delta ^ i (x-z) $, where $ZZ$ is the set of the zeros of $ psi$ , while $w(z) $ is the order of the generic zero $z $ and $C_(z,i) $ are arbitrary constants.
In the case of equation b ) , we have :
$psi = sinx $ , function which is analytic , not constant and therefore all its zeros are isolated , of finite and integer order: the zeros are all and only the points $x_k = kpi $ , with $ k in ZZ$ ; furtherly such $x_k $ are all simple zeros since :
$sin(x_k) = 0 $ , while $ (d sin(x_k))/dx =cos(x_k) ne 0 $ .
The theorem indicates that solutions to equation b) are all distributions $u(x) $ such that :
$u(x) = sum_(k=-oo)^(+oo)C_k delta(x-kpi)$ with $C_k $ being independent constants, since it is : $w(z) = 1,therefore i=0 $.
In addition, we note that for each selection of $C_k $ , the series representing $u(x) $ converges in the sense of distributions.
In a similar way we get the solutions of equations a ), c) :
a) $u(x) = c_1delta(x-2) +c_2 delta’(x-2) +c_3 delta(x-3) +c_4 delta’(x-3)$. $[x=2 ;x=3 $ are roots of order 2 ].
b) $u(x)=a_1 delta’(x)+a_2delta’’(x)+a_3 delta’(x-pi)+a_4delta’(x+pi)+sum_(k in ZZ) c_k(x-kpi)$.$[x=0 $ is a root of order 3 , while $x=+-pi$ are roots of order 2].
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