2021-Radhka, T. S. L. - Notes on Frobenius method

By T S L Radhika

BITS Pilani, Hyderabad Campus, Hyderabad INDIA

Download (PDF)

Licensed according to this deed.

Published on

Abstract

Radhka, T. S. L. 2021. Notes on Frobenius method

Abstract: This article explains and demonstrates applications of the Frobenius theory as discussed in the
book by Simmons on “Differential Equations with Applications and Historical Notes.” We specifically
discuss the existence of a second Frobenius solution to a linear second-order linear differential equation
when the two roots of its indicial equation differ by a positive integer and explain it with some examples.

Cite this work

Researchers should cite this work as follows:

  • T S L Radhika (2021), "2021-Radhka, T. S. L. - Notes on Frobenius method," https://simiode.org/resources/8785.

    BibTex | EndNote

Tags

Debug Console

Close11 MBRequestSessionTimelineDatabase50Events55
$_GET
Array( ) Array(

)
$_POST
Array( ) Array(

)
$_COOKIE
Array( ) Array(

)
$_SERVER
Array( REDIRECT_SCRIPT_URL => /resources/8785, REDIRECT_SCRIPT_URI => https://simiode.org/res... Array(
REDIRECT_SCRIPT_URL => /resources/8785,
REDIRECT_SCRIPT_URI => https://simiode.org/resources/8785,
REDIRECT_HTTPS => on,
REDIRECT_SSL_TLS_SNI => simiode.org,
REDIRECT_STATUS => 200,
SCRIPT_URL => /resources/8785,
SCRIPT_URI => https://simiode.org/resources/8785,
HTTPS => on,
SSL_TLS_SNI => simiode.org,
HTTP_USER_AGENT => CCBot/2.0 (https://commoncrawl.org/faq/),
HTTP_ACCEPT => text/html,application/xhtml+xml,application/xml;q=0.9,*/*;q=0.8,
HTTP_ACCEPT_LANGUAGE => en-US,en;q=0.5,
HTTP_ACCEPT_ENCODING => br,gzip,
HTTP_HOST => simiode.org,
HTTP_CONNECTION => Keep-Alive,
PATH => /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin,
SERVER_SIGNATURE => ,
SERVER_SOFTWARE => Apache/2.4.10 (Debian),
SERVER_NAME => simiode.org,
SERVER_ADDR => 74.208.170.104,
SERVER_PORT => 443,
REMOTE_ADDR => 3.238.95.208,
DOCUMENT_ROOT => /var/www/simiode,
REQUEST_SCHEME => https,
CONTEXT_PREFIX => ,
CONTEXT_DOCUMENT_ROOT => /var/www/simiode,
SERVER_ADMIN => [no address given],
SCRIPT_FILENAME => /var/www/simiode/index.php,
REMOTE_PORT => 58212,
REDIRECT_URL => /resources/8785,
GATEWAY_INTERFACE => CGI/1.1,
SERVER_PROTOCOL => HTTP/1.1,
REQUEST_METHOD => GET,
QUERY_STRING => ,
REQUEST_URI => /resources/8785,
SCRIPT_NAME => /index.php,
PHP_SELF => /index.php,
REQUEST_TIME_FLOAT => 1632451070.457,
REQUEST_TIME => 1632451070
)
separator .
orderBy id
orderDir asc
1 registerIP
2 access
always
renew
tracker.sid e2ddb4ac867ca78ada529e6713a63452
tracker.ssid e2ddb4ac867ca78ada529e6713a63452
tracker.rsid e2ddb4ac867ca78ada529e6713a63452
  • site 0.050 seconds (+0.050); 3.50 MB (+3.500) - afterInitialise
  • site 0.110 seconds (+0.060); 5.00 MB (+1.500) - afterRoute
  • site 0.374 seconds (+0.265); 10.75 MB (+5.750) - afterDispatch
  • site 0.386 seconds (+0.011); 10.75 MB (0.000) - beforeRenderModule mod_custom (Hub Footer)
  • site 0.388 seconds (+0.003); 10.75 MB (0.000) - afterRenderModule mod_custom (Hub Footer)
  • site 0.389 seconds (+0.000); 10.75 MB (0.000) - beforeRenderModule mod_googleanalytics (Google Analytics for Website)
  • site 0.391 seconds (+0.003); 10.75 MB (0.000) - afterRenderModule mod_googleanalytics (Google Analytics for Website)
  • site 0.392 seconds (+0.000); 10.75 MB (0.000) - beforeRenderModule mod_googleanalytics (Google Analytics 4 for Website)
  • site 0.392 seconds (+0.001); 10.75 MB (0.000) - afterRenderModule mod_googleanalytics (Google Analytics 4 for Website)
  • site 0.393 seconds (+0.000); 10.75 MB (0.000) - beforeRenderModule mod_collect (Collect)
  • site 0.395 seconds (+0.003); 11.00 MB (+0.250) - afterRenderModule mod_collect (Collect)
  • site 0.396 seconds (+0.001); 11.00 MB (0.000) - beforeRenderModule mod_breadcrumbs (Breadcrumbs)
  • site 0.405 seconds (+0.009); 11.00 MB (0.000) - afterRenderModule mod_breadcrumbs (Breadcrumbs)
  • site 0.405 seconds (+0.000); 11.00 MB (0.000) - beforeRenderModule mod_menu (SIMIODE Main Menu)
  • site 0.456 seconds (+0.051); 11.00 MB (0.000) - afterRenderModule mod_menu (SIMIODE Main Menu)
  • site 0.456 seconds (+0.000); 11.00 MB (0.000) - beforeRenderModule mod_search (Search)
  • site 0.460 seconds (+0.004); 11.00 MB (0.000) - afterRenderModule mod_search (Search)
  • site 0.468 seconds (+0.008); 11.00 MB (0.000) - onAfterRender
  • site 0.472 seconds (+0.004); 11.25 MB (+0.250) - sessionStore
11 MB

50 Queries Logged: 0.029477834701538 seconds

  1. SELECT data
      FROM `jos_session`
      WHERE `session_id` = '269905fcfb245228f3bf23809e4efc5e'
  2. DELETE
      FROM `jos_session`
      WHERE `session_id` = '269905fcfb245228f3bf23809e4efc5e'
  3. SELECT data
      FROM `jos_session`
      WHERE `session_id` = 'e2ddb4ac867ca78ada529e6713a63452'
  4. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_languages'
  5. SELECT session_id
      FROM `jos_session`
      WHERE `session_id` = 'e2ddb4ac867ca78ada529e6713a63452'
      LIMIT 1
  6. INSERT INTO `jos_session` (`session_id`,`client_id`,`time`,`ip`)
      VALUES ('e2ddb4ac867ca78ada529e6713a63452','0','1632451070','3.238.95.208')
  7. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_members'
  8. SELECT b.id
      FROM `jos_usergroups` AS `a`
      LEFT JOIN jos_usergroups AS b
      ON b.lft <= a.lft
      AND b.rgt >= a.rgt
      WHERE `a`.`id` = '1'
  9. SELECT *
      FROM `jos_viewlevels`
  10. SELECT folder AS type,element AS name,protected,params
      FROM `jos_extensions`
      WHERE `enabled` >= '1'
      AND `type` = 'plugin'
      AND `state` >= '0'
      AND `access` IN ('1','1','4')
      ORDER BY `ordering` ASC
  11. SELECT m.id,m.menutype,m.title,m.alias,m.note,m.path AS route,m.link,m.type,m.level,m.language,m.browserNav,m.access,m.params,m.home,m.img,m.template_style_id,m.component_id,m.parent_id,e.element AS component
      FROM `jos_menu` AS `m`
      LEFT JOIN jos_extensions AS e
      ON e.extension_id = m.component_id
      WHERE `m`.`published` = '1'
      AND `m`.`parent_id` > '0'
      AND `m`.`client_id` = '0'
      ORDER BY `m`.`lft` ASC
  12. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_resources'
  13. SELECT jos_template_styles.id,jos_template_styles.home,jos_template_styles.template,jos_template_styles.params,jos_extensions.protected
      FROM `jos_template_styles`
      INNER JOIN jos_extensions
      ON jos_extensions.element = jos_template_styles.template
      WHERE `jos_template_styles`.`client_id` = '0'
      AND `jos_extensions`.`enabled` = '1'
      AND `jos_extensions`.`type` = 'template'
      AND jos_extensions.`client_id` = `jos_template_styles`.`client_id`
      ORDER BY `home` DESC
  14. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_tags'
  15. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_tools'
  16. SELECT *
      FROM `jos_resources`
      WHERE `jos_resources`.`id` = '8785'
  17. SELECT *
      FROM `jos_resource_types`
      WHERE `jos_resource_types`.`id` = '64'
  18. SELECT jos_tags.*
      FROM `jos_tags`
      INNER JOIN jos_tags_object
      ON jos_tags.id = jos_tags_object.tagid
      WHERE `jos_tags_object`.`label` != 'badge'
      AND `jos_tags_object`.`tbl` = 'resources'
      AND `jos_tags_object`.`objectid` = '8785'
      AND `jos_tags`.`admin` = '0'
  19. SELECT *
      FROM `jos_author_assoc`
      WHERE `jos_author_assoc`.`subid` = '8785'
      AND `jos_author_assoc`.`subtable` = 'resources'
      ORDER BY `ordering` ASC
  20. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_citations'
  21. SHOW TABLES
  22. SELECT *
      FROM `jos_resource_stats`
      WHERE `jos_resource_stats`.`resid` = '8785'
      AND `jos_resource_stats`.`period` = '14'
      AND `jos_resource_stats`.`datetime` LIKE '%2021-09-%%'
  23. SELECT *
      FROM `jos_resource_stats_clusters`
      WHERE `jos_resource_stats_clusters`.`resid` = '8785'
  24. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_publications'
  25. SELECT *
      FROM `jos_citations`
      INNER JOIN jos_citations_assoc
      ON jos_citations_assoc.cid = jos_citations.id
      WHERE `jos_citations`.`published` = '1'
      AND `jos_citations_assoc`.`tbl` = 'resource'
      AND `jos_citations_assoc`.`oid` = '8785'
      ORDER BY `jos_citations`.`affiliated` ASC,`jos_citations`.`year` DESC
  26. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_answers'
  27. SELECT jos_answers_questions.*
      FROM `jos_answers_questions`
      INNER JOIN jos_tags_object
      ON jos_tags_object.objectid = jos_answers_questions.id
      INNER JOIN jos_tags
      ON jos_tags.id = jos_tags_object.tagid
      WHERE `jos_tags_object`.`tbl` = 'answers'
      AND `jos_tags`.`tag` IN ('resource8785')
      AND `jos_answers_questions`.`state` < '2'
  28. SELECT *
      FROM `jos_users`
      WHERE `jos_users`.`id` = '-1582826630'
  29. SHOW FULL COLUMNS
      FROM `jos_resource_assoc`
  30. SELECT jos_resources.*,parent_id,jos_resource_assoc.ordering AS associative_ordering,jos_resource_assoc.grouping AS associative_grouping
      FROM `jos_resources`
      INNER JOIN jos_resource_assoc
      ON jos_resources.id = child_id
      WHERE `jos_resource_assoc`.`parent_id` = '8785'
      AND `jos_resources`.`standalone` = '0'
      AND `jos_resources`.`published` = '1'
      ORDER BY `ordering` ASC
  31. SELECT *
      FROM `jos_resource_types`
      WHERE `jos_resource_types`.`id` = '0'
  32. SELECT *
      FROM `jos_resource_types`
      WHERE `jos_resource_types`.`id` = '33'
  33. SELECT *
      FROM `jos_resource_licenses`
      WHERE `jos_resource_licenses`.`name` = 'cc'
      LIMIT 1
  34. SELECT k.*
      FROM ((SELECT DISTINCT w.id, w.title, w.pagename AS alias, v.pagetext AS introtext, NULL AS type, NULL AS published, NULL AS publish_up, w.scope, w.rating, w.times_rated, w.ranking, 'Topic' AS section
      FROM `jos_wiki_pages` AS w
      INNER JOIN `jos_wiki_versions` AS v
      ON w.version_id=v.id
      INNER JOIN `jos_wiki_links` AS wl
      ON wl.page_id=w.id
      WHERE v.approved=1
      AND wl.scope='resource'
      AND wl.scope_id='8785'
      AND w.access!=1
      GROUP BY w.id, v.pagetext
      ORDER BY ranking DESC, title
      LIMIT 10) UNION (SELECT DISTINCT r.id, r.title, r.alias, r.introtext, r.type, r.published, r.publish_up, NULL AS scope, r.rating, r.times_rated, r.ranking, rt.type AS section
      FROM `jos_resource_types` AS rt, `jos_resources` AS r JOIN `jos_resource_assoc` AS a
      ON r.id=a.parent_id
      LEFT JOIN `jos_resource_types` AS t
      ON r.logical_type=t.id
      WHERE r.published=1
      AND a.child_id='8785'
      AND r.type=rt.id
      AND r.type!=8
      AND r.access=0
      ORDER BY r.ranking
      LIMIT 10)) AS k
      ORDER BY ranking DESC
      LIMIT 10
  35. SELECT *
      FROM `jos_resource_sponsors`
      WHERE `jos_resource_sponsors`.`state` = '1'
  36. SELECT jos_tags_object.label,jos_tags.*
      FROM `jos_tags`
      INNER JOIN jos_tags_object
      ON jos_tags_object.tagid = jos_tags.id
      WHERE `jos_tags_object`.`tbl` = 'resources'
      AND `jos_tags_object`.`objectid` = '8785'
      GROUP BY jos_tags_object.label,jos_tags.id
  37. SELECT *
      FROM `jos_resource_types`
      WHERE `jos_resource_types`.`id` = '64'
  38. SELECT m.id,m.title,m.module,m.position,m.content,m.showtitle,m.params,mm.menuid,e.protected
      FROM `jos_modules` AS `m`
      LEFT JOIN jos_modules_menu AS mm
      ON mm.moduleid = m.id
      LEFT JOIN jos_extensions AS e
      ON e.element = m.module
      AND e.client_id = m.client_id
      WHERE `m`.`published` = '1'
      AND `e`.`enabled` = '1'
      AND (`m`.`publish_up` IS NULL OR `m`.`publish_up` <= '2021-09-24 02:37:50' )
      AND (`m`.`publish_down` IS NULL OR `m`.`publish_down` >= '2021-09-24 02:37:50' )
      AND `m`.`access` IN ('1','1','4')
      AND `m`.`client_id` = '0'
      AND (`mm`.`menuid` = '88' OR `mm`.`menuid` <= '0' )
      ORDER BY `m`.`position` ASC,`m`.`ordering` ASC
  39. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_search'
  40. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_users'
  41. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_content'
  42. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_categories'
  43. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_whatsnew'
  44. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_newsletter'
  45. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_blog'
  46. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_support'
  47. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_kb'
  48. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_groups'
  49. SELECT extension_id AS id,element AS "option",params,enabled
      FROM `jos_extensions`
      WHERE `type` = 'component'
      AND `element` = 'com_forum'
  50. UPDATE `jos_session`
      SET `data` = '__default|a:9:{s:15:\"session.counter\";i:1;s:19:\"session.timer.start\";i:1632451070;s:18:\"session.timer.last\";i:1632451070;s:17:\"session.timer.now\";i:1632451070;s:8:\"registry\";O:23:\"Hubzero\\Config\\Registry\":2:{s:7:\"\0*\0data\";O:8:\"stdClass\":2:{s:1:\"0\";s:7:\"session\";s:10:\"Components\";O:8:\"stdClass\":1:{s:9:\"Resources\";O:8:\"stdClass\":1:{s:6:\"Models\";O:8:\"stdClass\":1:{s:6:\"Author\";O:8:\"stdClass\":2:{s:7:\"orderby\";s:8:\"ordering\";s:8:\"orderdir\";s:3:\"asc\";}}}}}s:9:\"separator\";s:1:\".\";}s:4:\"user\";C:17:\"Hubzero\\User\\User\":22:{a:1:{s:5:\"guest\";b:1;}}s:11:\"tracker.sid\";s:32:\"e2ddb4ac867ca78ada529e6713a63452\";s:12:\"tracker.ssid\";s:32:\"e2ddb4ac867ca78ada529e6713a63452\";s:12:\"tracker.rsid\";s:32:\"e2ddb4ac867ca78ada529e6713a63452\";}',`time` = '1632451070',`ip` = '3.238.95.208'
      WHERE `session_id` = 'e2ddb4ac867ca78ada529e6713a63452'
  • onAfterInitialise Method plgSystemP3p::onAfterInitialise /core/plugins/system/p3p/p3p.php:21
  • onAfterInitialise Method plgSystemRemember::onAfterInitialise /core/plugins/system/remember/remember.php:20
  • onAfterInitialise Method plgSystemHubzero::onAfterInitialise /core/plugins/system/hubzero/hubzero.php:21
  • onAfterInitialise Method plgSystemXFeed::onAfterInitialise /core/plugins/system/xfeed/xfeed.php:21
  • onAfterInitialise Method plgSystemSupergroup::onAfterInitialise /core/plugins/system/supergroup/supergroup.php:113
  • onAfterInitialise Method plgSystemReferrerpolicy::onAfterInitialise /core/plugins/system/referrerpolicy/referrerpolicy.php:21
  • onAfterRoute Method plgSystemJquery::onAfterRoute /core/plugins/system/jquery/jquery.php:21
  • onAfterRoute Method plgSystemSpamjail::onAfterRoute /core/plugins/system/spamjail/spamjail.php:21
  • onAfterRoute Method plgSystemIncomplete::onAfterRoute /core/plugins/system/incomplete/incomplete.php:21
  • onAfterRoute Method plgSystemUnconfirmed::onAfterRoute /core/plugins/system/unconfirmed/unconfirmed.php:21
  • onAfterRoute Method plgSystemUnapproved::onAfterRoute /core/plugins/system/unapproved/unapproved.php:21
  • onAfterRoute Method plgSystemPassword::onAfterRoute /core/plugins/system/password/password.php:21
  • onResourcesAreas Method plgResourcesAbout::onResourcesAreas /core/plugins/resources/about/about.php:32
  • onResourcesAreas Method plgResourcesSupportingDocs::onResourcesAreas /core/plugins/resources/supportingdocs/supportingdocs.php:30
  • onResourcesAreas Method plgResourcesReviews::onResourcesAreas /core/plugins/resources/reviews/reviews.php:43
  • onResourcesAreas Method plgResourcesUsage::onResourcesAreas /core/plugins/resources/usage/usage.php:29
  • onResourcesAreas Method plgResourcesVersions::onResourcesAreas /core/plugins/resources/versions/versions.php:29
  • onResourcesAreas Method plgResourcesShare::onResourcesAreas /core/plugins/resources/share/share.php:29
  • onResourcesAreas Method plgResourcesCitations::onResourcesAreas /core/plugins/resources/citations/citations.php:29
  • onResourcesAreas Method plgResourcesQuestions::onResourcesAreas /core/plugins/resources/questions/questions.php:29
  • onResourcesAreas Method plgResourcesWishlist::onResourcesAreas /core/plugins/resources/wishlist/wishlist.php:29
  • onResourcesAreas Method plgResourcesFindThisText::onResourcesAreas /core/plugins/resources/findthistext/findthistext.php:29
  • onResourcesAreas Method plgResourcesWindowstools::onResourcesAreas /core/plugins/resources/windowstools/windowstools.php:214
  • onResourcesAreas Method plgResourcesComments::onResourcesAreas /app/plugins/resources/comments/comments.php:61
  • onContentPrepare Method plgContentLoadmodule::onContentPrepare /core/plugins/content/loadmodule/loadmodule.php:35
  • onContentPrepare Method plgContentPagebreak::onContentPrepare /core/plugins/content/pagebreak/pagebreak.php:35
  • onContentPrepare Method plgContentXhubtags::onContentPrepare /core/plugins/content/xhubtags/xhubtags.php:25
  • onContentPrepare Method plgContentFormatwiki::onContentPrepare /core/plugins/content/formatwiki/formatwiki.php:63
  • onContentPrepare Method plgContentFormathtml::onContentPrepare /core/plugins/content/formathtml/formathtml.php:105
  • onContentPrepare Method plgContentExternalhref::onContentPrepare /core/plugins/content/externalhref/externalhref.php:41
  • onResources Method plgResourcesAbout::onResources /core/plugins/resources/about/about.php:54
  • onResources Method plgResourcesSupportingDocs::onResources /core/plugins/resources/supportingdocs/supportingdocs.php:52
  • onResources Method plgResourcesReviews::onResources /core/plugins/resources/reviews/reviews.php:91
  • onResources Method plgResourcesUsage::onResources /core/plugins/resources/usage/usage.php:51
  • onResources Method plgResourcesVersions::onResources /core/plugins/resources/versions/versions.php:50
  • onResources Method plgResourcesShare::onResources /core/plugins/resources/share/share.php:50
  • onResources Method plgResourcesCitations::onResources /core/plugins/resources/citations/citations.php:51
  • onResources Method plgResourcesQuestions::onResources /core/plugins/resources/questions/questions.php:60
  • onResources Method plgResourcesWishlist::onResources /core/plugins/resources/wishlist/wishlist.php:50
  • onResources Method plgResourcesFindThisText::onResources /core/plugins/resources/findthistext/findthistext.php:51
  • onResources Method plgResourcesWindowstools::onResources /core/plugins/resources/windowstools/windowstools.php:235
  • onResources Method plgResourcesGooglescholar::onResources /core/plugins/resources/googlescholar/googlescholar.php:25
  • onResources Method plgResourcesOpengraph::onResources /core/plugins/resources/opengraph/opengraph.php:25
  • onResources Method plgResourcesDublincore::onResources /core/plugins/resources/dublincore/dublincore.php:25
  • onResources Method plgResourcesComments::onResources /app/plugins/resources/comments/comments.php:82
  • onResources Method plgResourcesCoins::onResources /core/plugins/resources/coins/coins.php:25
  • onResourcesSub Method plgResourcesGroups::onResourcesSub /core/plugins/resources/groups/groups.php:45
  • onResourcesSub Method plgResourcesRelated::onResourcesSub /core/plugins/resources/related/related.php:45
  • onResourcesSub Method plgResourcesSponsors::onResourcesSub /core/plugins/resources/sponsors/sponsors.php:45
  • onResourcesSub Method plgResourcesWatch::onResourcesSub /core/plugins/resources/watch/watch.php:46
  • onResourcesSub Method plgResourcesCollections::onResourcesSub /core/plugins/resources/collections/collections.php:31
  • onAfterDispatch Method plgSystemMobile::onAfterDispatch /core/plugins/system/mobile/mobile.php:22
  • onAfterDispatch Method plgSystemDebug::onAfterDispatch /core/plugins/system/debug/debug.php:60
  • onAfterDispatch Method PlgSystemHighlight::onAfterDispatch /core/plugins/system/highlight/highlight.php:25
  • onAfterRender Method plgSystemSef::onAfterRender /core/plugins/system/sef/sef.php:21