Value of $f(0)$ in differential equation.











up vote
1
down vote

favorite













If $f(pi) = 2$ and $displaystyle int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx=5.$



Then value of $f(0).$



Given that function $f(x)$ is continuous in $[0,pi]$




Try: $$int bigg[f(x)+f''(x)bigg]sin xdx$$



$$ = int f(x)sin xdx+sin xf'(x)+int cos xf'(x)dx$$



$$ = int f(x)sin xdx+sin xf'(x)+cos xf(x)-int sin x f(x)dx$$



So $$int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx = sin xf'(x)+cos x f(x) = 5.$$



So $$sin xf'(x)+cos xf(x) = 5$$



Could some help me how i solve above differential equation. Thanks






differential-equations differential







share|cite|improve this question


















  • 1




    You should have done with the limits.
    – Yadati Kiran
    Nov 22 at 9:56















up vote
1
down vote

favorite













If $f(pi) = 2$ and $displaystyle int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx=5.$



Then value of $f(0).$



Given that function $f(x)$ is continuous in $[0,pi]$




Try: $$int bigg[f(x)+f''(x)bigg]sin xdx$$



$$ = int f(x)sin xdx+sin xf'(x)+int cos xf'(x)dx$$



$$ = int f(x)sin xdx+sin xf'(x)+cos xf(x)-int sin x f(x)dx$$



So $$int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx = sin xf'(x)+cos x f(x) = 5.$$



So $$sin xf'(x)+cos xf(x) = 5$$



Could some help me how i solve above differential equation. Thanks






differential-equations differential







share|cite|improve this question


















  • 1




    You should have done with the limits.
    – Yadati Kiran
    Nov 22 at 9:56













up vote
1
down vote

favorite









up vote
1
down vote

favorite












If $f(pi) = 2$ and $displaystyle int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx=5.$



Then value of $f(0).$



Given that function $f(x)$ is continuous in $[0,pi]$




Try: $$int bigg[f(x)+f''(x)bigg]sin xdx$$



$$ = int f(x)sin xdx+sin xf'(x)+int cos xf'(x)dx$$



$$ = int f(x)sin xdx+sin xf'(x)+cos xf(x)-int sin x f(x)dx$$



So $$int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx = sin xf'(x)+cos x f(x) = 5.$$



So $$sin xf'(x)+cos xf(x) = 5$$



Could some help me how i solve above differential equation. Thanks






differential-equations differential







share|cite|improve this question














If $f(pi) = 2$ and $displaystyle int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx=5.$



Then value of $f(0).$



Given that function $f(x)$ is continuous in $[0,pi]$




Try: $$int bigg[f(x)+f''(x)bigg]sin xdx$$



$$ = int f(x)sin xdx+sin xf'(x)+int cos xf'(x)dx$$



$$ = int f(x)sin xdx+sin xf'(x)+cos xf(x)-int sin x f(x)dx$$



So $$int^{pi}_{0}bigg[f(x)+f''(x)bigg]sin(x)dx = sin xf'(x)+cos x f(x) = 5.$$



So $$sin xf'(x)+cos xf(x) = 5$$



Could some help me how i solve above differential equation. Thanks






differential-equations differential




differential-equations differential






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 22 at 9:44









Durgesh Tiwari

5,2102628




5,2102628








  • 1




    You should have done with the limits.
    – Yadati Kiran
    Nov 22 at 9:56














  • 1




    You should have done with the limits.
    – Yadati Kiran
    Nov 22 at 9:56








1




1




You should have done with the limits.
– Yadati Kiran
Nov 22 at 9:56




You should have done with the limits.
– Yadati Kiran
Nov 22 at 9:56










2 Answers
2






active

oldest

votes

















up vote
3
down vote



accepted










Integrate by part twice. $int_0^{pi} f''(x)sin, x , dx= -int_0^{pi} f'(x)cos, x , dx=f(0) -f(pi)(-1)-int_0^{pi} f(x)sin, x , dx$ so we get $5=f(0) +f(pi)$. Hence $f(0)=3$.






share|cite|improve this answer




























    up vote
    2
    down vote













    In fact, you have obtained that $$f'(x)sin x+f(x)cos x|_{0}^{pi}=5$$which means that $$-f(pi)+f(0)=5$$from which we obtain$$f(0)=7$$






    share|cite|improve this answer





















      Your Answer





      StackExchange.ifUsing("editor", function () {
      return StackExchange.using("mathjaxEditing", function () {
      StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
      StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
      });
      });
      }, "mathjax-editing");

      StackExchange.ready(function() {
      var channelOptions = {
      tags: "".split(" "),
      id: "69"
      };
      initTagRenderer("".split(" "), "".split(" "), channelOptions);

      StackExchange.using("externalEditor", function() {
      // Have to fire editor after snippets, if snippets enabled
      if (StackExchange.settings.snippets.snippetsEnabled) {
      StackExchange.using("snippets", function() {
      createEditor();
      });
      }
      else {
      createEditor();
      }
      });

      function createEditor() {
      StackExchange.prepareEditor({
      heartbeatType: 'answer',
      convertImagesToLinks: true,
      noModals: true,
      showLowRepImageUploadWarning: true,
      reputationToPostImages: 10,
      bindNavPrevention: true,
      postfix: "",
      imageUploader: {
      brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
      contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
      allowUrls: true
      },
      noCode: true, onDemand: true,
      discardSelector: ".discard-answer"
      ,immediatelyShowMarkdownHelp:true
      });


      }
      });














       

      draft saved


      draft discarded


















      StackExchange.ready(
      function () {
      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008922%2fvalue-of-f0-in-differential-equation%23new-answer', 'question_page');
      }
      );

      Post as a guest















      Required, but never shown

























      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes








      up vote
      3
      down vote



      accepted










      Integrate by part twice. $int_0^{pi} f''(x)sin, x , dx= -int_0^{pi} f'(x)cos, x , dx=f(0) -f(pi)(-1)-int_0^{pi} f(x)sin, x , dx$ so we get $5=f(0) +f(pi)$. Hence $f(0)=3$.






      share|cite|improve this answer

























        up vote
        3
        down vote



        accepted










        Integrate by part twice. $int_0^{pi} f''(x)sin, x , dx= -int_0^{pi} f'(x)cos, x , dx=f(0) -f(pi)(-1)-int_0^{pi} f(x)sin, x , dx$ so we get $5=f(0) +f(pi)$. Hence $f(0)=3$.






        share|cite|improve this answer























          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          Integrate by part twice. $int_0^{pi} f''(x)sin, x , dx= -int_0^{pi} f'(x)cos, x , dx=f(0) -f(pi)(-1)-int_0^{pi} f(x)sin, x , dx$ so we get $5=f(0) +f(pi)$. Hence $f(0)=3$.






          share|cite|improve this answer












          Integrate by part twice. $int_0^{pi} f''(x)sin, x , dx= -int_0^{pi} f'(x)cos, x , dx=f(0) -f(pi)(-1)-int_0^{pi} f(x)sin, x , dx$ so we get $5=f(0) +f(pi)$. Hence $f(0)=3$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 22 at 9:51









          Kavi Rama Murthy

          42k31751




          42k31751






















              up vote
              2
              down vote













              In fact, you have obtained that $$f'(x)sin x+f(x)cos x|_{0}^{pi}=5$$which means that $$-f(pi)+f(0)=5$$from which we obtain$$f(0)=7$$






              share|cite|improve this answer

























                up vote
                2
                down vote













                In fact, you have obtained that $$f'(x)sin x+f(x)cos x|_{0}^{pi}=5$$which means that $$-f(pi)+f(0)=5$$from which we obtain$$f(0)=7$$






                share|cite|improve this answer























                  up vote
                  2
                  down vote










                  up vote
                  2
                  down vote









                  In fact, you have obtained that $$f'(x)sin x+f(x)cos x|_{0}^{pi}=5$$which means that $$-f(pi)+f(0)=5$$from which we obtain$$f(0)=7$$






                  share|cite|improve this answer












                  In fact, you have obtained that $$f'(x)sin x+f(x)cos x|_{0}^{pi}=5$$which means that $$-f(pi)+f(0)=5$$from which we obtain$$f(0)=7$$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Nov 22 at 10:16









                  Mostafa Ayaz

                  12.3k3733




                  12.3k3733






























                       

                      draft saved


                      draft discarded



















































                       


                      draft saved


                      draft discarded














                      StackExchange.ready(
                      function () {
                      StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008922%2fvalue-of-f0-in-differential-equation%23new-answer', 'question_page');
                      }
                      );

                      Post as a guest















                      Required, but never shown





















































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown

































                      Required, but never shown














                      Required, but never shown












                      Required, but never shown







                      Required, but never shown







                      -

                      Popular posts from this blog

                      Berounka

                      Image
                      Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc...
                      Read more

                      Sphinx de Gizeh

                      Image
                      Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
                      Read more

                      Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

                      Image
                      3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
                      Read more