text: # Linear Differential Operator
  
  Let $\phi({D})$ be a differential operator defined by $$D^n+a_{n-1}D^{n-1}+a_{n-2}D^{n-2}+...+a_0=\sum_{j=0}^{n}a_jD^j$$
  
  with constant coefficients $a_j$ (setting $a_n=1$) and the derivative operator $D=\frac{d}{dx}$. Let $f(x)$ be an arbitrarily often differentiable function. Show that:
  
  1. $\phi(D)[e^{\lambda x}f(x)]=e^{\lambda x} \phi(D+\lambda)f(x)$
  
  2. $\phi(D)[x f(x)]=x \phi(D)f(x)+\phi'(D)f(x)$
  
  It's relatively easy to show that this is true for the first couple of terms but I don' know how to show it for all $n$. Is there some elegant quick way to show that 1. and 2. are true?
  
  In both cases first verify the claim for $\phi(D)=D,$ then by induction for $\phi(D)=D^n$ (this is the most interesting step), then by linearity for a general polynomial $\phi.$
  Or if you are as lazy as I am, your first step could even be the constant $\phi(D)=1.$
explanation: The document presents a mathematical problem involving a linear differential operator \(\phi(D)\). It defines the operator and asks the reader to show two properties of this operator:
  
  1. \(\phi(D)[e^{\lambda x}f(x)]=e^{\lambda x} \phi(D+\lambda)f(x)\)
  2. \(\phi(D)[x f(x)]=x \phi(D)f(x)+\phi'(D)f(x)\)
  
  The document also provides a hint on how to approach the proof, suggesting to verify the claims for \(\phi(D)=D\), then by induction for \(\phi(D)=D^n\), and finally by linearity for a general polynomial \(\phi\).
  
  The document contains a question but does not provide an answer to it.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Linear Differential Operator
  
  Let $\phi({D})$ be a differential operator defined by $$D^n+a_{n-1}D^{n-1}+a_{n-2}D^{n-2}+...+a_0=\sum_{j=0}^{n}a_jD^j$$
  
  with constant coefficients $a_j$ (setting $a_n=1$) and the derivative operator $D=\frac{d}{dx}$. Let $f(x)$ be an arbitrarily often differentiable function. Show that:
  
  1. $\phi(D)[e^{\lambda x}f(x)]=e^{\lambda x} \phi(D+\lambda)f(x)$
  
  2. $\phi(D)[x f(x)]=x \phi(D)f(x)+\phi'(D)f(x)$
  
  It's relatively easy to show that this is true for the first couple of terms but I don' know how to show it for all $n$. Is there some elegant quick way to show that 1. and 2. are true?
  
  In both cases first verify the claim for $\phi(D)=D,$ then by induction for $\phi(D)=D^n$ (this is the most interesting step), then by linearity for a general polynomial $\phi.$
  Or if you are as lazy as I am, your first step could even be the constant $\phi(D)=1.$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document presents a mathematical problem involving a linear differential operator \(\phi(D)\). It defines the operator and asks the reader to show two properties of this operator:
  
  1. \(\phi(D)[e^{\lambda x}f(x)]=e^{\lambda x} \phi(D+\lambda)f(x)\)
  2. \(\phi(D)[x f(x)]=x \phi(D)f(x)+\phi'(D)f(x)\)
  
  The document also provides a hint on how to approach the proof, suggesting to verify the claims for \(\phi(D)=D\), then by induction for \(\phi(D)=D^n\), and finally by linearity for a general polynomial \(\phi\).
  
  The document contains a question but does not provide an answer to it.
  
  Label: question