23.5 Linear equations (with constant coefficients)

Definition 438

[23Z]We formally indicate with \(D\) the operation ”computing of the derivative”. Given a polynomial \(p(x)\)

\[ p(x)=a_ n x^{n} + a_{n-1} x^{n-1} + \ldots + a_{1} x + a_ 0 \]

(which has constants coefficients \(a_ i∈ℂ\)) we formally construct the linear operator

\[ p(D)=a_ n D^{n} + a_{n-1} D^{n-1} + \dots a_{1} D + a_ 0 \]

which transforms a function \(f:ℝ→ℂ\) of class \(C^{n+k}\) into the function \(p(D) f\), class at least \(C^ k\), defined pointwise by

\[ [p(D) f] (x) {\stackrel{.}{=}}a_ n f^{(n)}(x) + a_{n-1} f^{(n-1)}(x) + \dots a_{1} f'(x) + a_ 0 f(x)\quad . \]

E438

[1SC] Given two polynomials \(p(x),q(x)\) and the product polynomial \(r(x)=p(x)q(x)\), show that \(p(D) [q(D) f ] = r(D) f\)

E438

[1SD] Define \(f(x) = e^{𝜆 x}\), note that

\[ [p(D) f ](x) = p(𝜆) f(x) \quad . \]

We can therefore consider exponentials \(e^{𝜆 x}\) as eigenvectors of \(p(D)\), with eigenvalue \(p(𝜆)\).

E438

[1SF] Let \(f:ℝ→ℂ\) be a \(C^ n\) class function , let \(𝜃∈ℂ\) be a constant, and let \(g(x)= e^{𝜃 x}f(x)\). Show that, if \(p\) is a polynomial and \(q(x)=p(x+𝜃)\), then

\[ p(D) g = e^{𝜃 x} [q(D) f] \quad . \]

Note that we can also write the relation above as a ”conjugation”

\[ e^{-𝜃 x} \big[p(D) [ e^{𝜃 x} f ]\big] = p(D+𝜃) f ~ . \]

Hidden solution: [UNACCESSIBLE UUID ’1SG’]

E438

[1SH] Prerequisites:3.Given \(𝜃∈ℂ\) and \(k∈ℕ\), define \(p(x)=(x-𝜃)^ k\), show that \(p(D)f=0\) if and only if \(f(x)=e^{𝜃 x}r(x) \) with \(r\) polynomial of degree at most \(k-1\).

Hidden solution: [UNACCESSIBLE UUID ’1SJ’]

E438

[1SK] Prerequisites:1, 3.

Fix \(𝜃,𝜏∈ℂ\) with \(𝜃≠ 𝜏\), \(q(x)\) a polynomial, and \(k∈ℕ\). Let’s define \(p(x)=(x-𝜃)^ k\). Show that

\[ p(D)f(x)=e^{𝜏 x}q(x) \]

if and only if

\[ f(x)=e^{𝜃 x} r(x) + e^{𝜏 x} \tilde q(x) ~ , \]

with \(r\) polynomial of degree at most \(k-1\) and \(\tilde q\) polynomial of the same degree as \(q\).

Hidden solution: [UNACCESSIBLE UUID ’1SM’]

E438

[1SN]Given \(a_ 0\ldots a_{n}∈ℂ\) constants, with \(a_ n≠ 0\), and defining \(p(x)=a_ n x^{n} + a_{n-1} x^{n-1} + \dots a_{1} x + a_ 0\), describe all possible solutions \(f\) of

\[ p(D) f = 0~ . \]

Show that the solution space is a vector space (based on the field \(ℂ\) of complex numbers) of dimension \(n\).

( Hint. Factorize the polynomial and take advantage of previous exercises. ).

E438

[1SP]Prerequisites:6.With \(p\) as above, also analyze the problem

\[ p(D) f = e^{𝛼 x} \]

(with \(𝛼∈ℂ\) constant).

What happens when \(𝛼\) approaches a root of the polynomial \(p\)?

[UNACCESSIBLE UUID ’1SQ’][1SR]Given parameters \(y_ 0,\ldots ,y_{n-1}∈ℂ\), and also \(𝛼∈ℂ\), the solution of the Cauchy problem

\[ \begin{cases} p(D) f = e^{𝛼 x} \\ f(0)=y_ 0,\\ \ldots \\ f^{n-1}(0)=y_{n-1} \end{cases} \]

exists for all times, and depends continuously on the parameters \(𝛼,y_ 0,\ldots ,y_{n-1}∈ℂ\).

E438

[1SS]Given \(h=h(x)\), and \(𝜃∈ℝ\), solve the differential equations

\[ (D-𝜃) f (x) = h (x) \]
\[ (D-𝜃)^ 2 f (x) = h (x) \]
\[ (D^ 2+𝜃^ 2) f (x) = h (x) \]
\[ (D^ 2-𝜃^ 2) f (x) = h (x) \]

and special cases

\[ (D-1) f (x) = x^ k \]
\[ (D-𝜃) f (x) = e^{𝛼 x} \]

(with \(𝛼∈ℂ\), and \(k∈ℕ\), constants).

Hidden solution: [UNACCESSIBLE UUID ’1SV’]

[UNACCESSIBLE UUID ’1ST’]