Powershell: removing Xml element failed

I've got an xml file, containing several book names:

<?xml version="1.0" encoding="utf-8"?>
<Project>
  <ItemGroup>
    <Book Name="Learn Powershell" />
    <Book Name=".net Programming" />
    <Book Name="C# Programming" />
  </ItemGroup>
</Project>

Then I've got a powershell script, trying to find out book name that containing "powershell" and remove it from xml:

[xml]$xml=Get-Content "D:\m.xml"
$p = $xml.Project.ItemGroup.Book
$node=$p|?{$_.Name.Contains("Powershell")}
$node.Parent.RemoveChilde($node)

It reports runtime exception:

PS D:\> D:\Untitled4.ps1
You cannot call a method on a null-valued expression.
At D:\Untitled4.ps1:4 char:1
+ $node.Parent.RemoveChild($node)
+ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ CategoryInfo          : InvalidOperation: (:) [], RuntimeException
+ FullyQualifiedErrorId : InvokeMethodOnNull

Why? How to correct my code?