Applying Site Page as default home page in SharePoint 2010 and 2013

1. Programatically......Changing the Welcome  Page  

public static void ChangeWelcomePage(SPWeb rootWeb, string pageName)
{
    if (PublishingWeb.IsPublishingWeb(rootWeb))
    {
  PublishingWeb pubWeb = PublishingWeb.GetPublishingWeb(rootWeb);
  //Get the file name
  SPFile welcomeFile = rootWeb.GetFile(pageName);
  //Assign the new filename to the DefaultPage property
  pubWeb.DefaultPage = welcomeFile;
  //Update the Publishing Web.
  pubWeb.Update();
    }
}

Power Shell .

Remove-PSSnapin Microsoft.SharePoint.PowerShell -erroraction SilentlyContinue
Add-PSSnapin Microsoft.SharePoint.PowerShell -erroraction SilentlyContinue

################################################### code###################################################

$sourceWeb = $null
try
{
$SiteUrl = "http://adigroup/myteamsite"
$sourceWeb = Get-SPSite -identity $SiteUrl
$oFolder = $sourceWeb.RootFolder;
$oFolder.WelcomePage = "Pages/MyCustomPage.aspx";
$oFolder.Update();

}
catch
{
write-host $_.exception

}
finally
{
if($sourceWeb -ne $null){$sourceWeb.Dispose()}

}