With no other term, the equations are called homogeneous equations. 9�� (�( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��( ��itX~t �)�D?�? (�� (�� (�� (�� equations. But in … Find the eigenvalues and eigenvectors of the matrix Answer. The columns of a Markov matrix add to 1 but in the differential equation situation, they'll add to 0. Fitting the linear combination to the initial conditions, you get a real solution of the differential equation. Section 5-7 : Real Eigenvalues. Solution: Find the eigenvalues first. This might introduce extra solutions. (�� (1) We say an eigenvalue λ 1 of A is repeated if it is a multiple root of the char­ acteristic equation of A; in our case, as this is a quadratic equation, the only possible case is when λ 1 is a double real root. [�ը�:��B;Y�9o�z�]��(�#sz��EQ�QE QL�X�v�M~ǈ�� ^y5˰Q�T��;D�����y�s��U�m"��noS@������ժ�6QG�|��Vj��o��P��\� V[���0\�� (�� It’s now time to start solving systems of differential equations. endstream xڍ�;O�0��� Systems of Differential Equations with Zero Eigenvalues are investigated. P�NA��R"T��Т��p��� �Zw0qkp��)�(�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (�� (4Q@Q@#0U,{R�M��I�*��f%����E��QE QE %Q@>9Z>��Je���c�d����+:������R�c*}�TR+S�KVdQE QE QE QE QE QE QE QE QE QE QE QE QE QE QE w�� (�� *��̧ۊ�Td9���L�)�6�(��(��(��(���( ��(U�T�Gp��pj�ӱ2���ER�f���ҭG"�>Sϥh��e�QE2�(��(��(��(��(��(��(��(��(��(��(��(��(��( QE t��rsW�8���Q���0��* B�(��(��(���� J(�� ��(�� (�� (�� 4 0 obj �� � } !1AQa"q2���#B��R��$3br� (�� /Filter /FlateDecode (�� In the last section, we found that if x' = Ax. endobj � QE �p��U�)�M��u�ͩ���T� EPEPEP0��(��er0X�(��Z�EP0��( ��( ��( ��cȫ�'ژ7a�֑W��*-�H�P���3s)�=Z�'S�\��p���SEc#�!�?Z�1�0��>��2ror(���>��KE�QP�s?y�}Z ���x�;s�ިIy4�lch>�i�X��t�o�h ��G;b]�����YN� P}z�蠎!�/>��J �#�|��S֤�� (�� (�� (�� (�� J(4PEPW}MU�G�QU�9noO��*K endstream 28 0 obj << :wZ�EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPE� QE QE TR��ɦ�K��^��K��! ���� JFIF   �� C 2 0 obj Using Euler's formula , the solutions take the form . Solving DE systems with complex eigenvalues. �ph��,Gs�� :�# �Vu9$d? (�� Solutions to Systems – We will take a look at what is involved in solving a system of differential equations. SAMPLE APPLICATION OF DIFFERENTIAL EQUATIONS 3 Sometimes in attempting to solve a de, we might perform an irreversible step. is a solution. It’s now time to start solving systems of differential equations. Finding solutions when there are complex eigenvalues is considerably more difﬁcult. →x = →η eλt x → = η → e λ t. where λ λ and →η η → are eigenvalues and eigenvectors of the matrix A A. Solving deconstructed matrix ordinary differential equations. 36 0 obj << (Note that x and z are vectors.) x = Ax. Systems of Differential Equations – Here we will look at some of the basics of systems of differential equations. Systems with Complex Eigenvalues. (�� We will use this identity when solving systems of differential equations with constant coefficients in which the eigenvalues are complex. So there is the eigenvalue of 1 for our powers is like the eigenvalue 0 for differential equations. %PDF-1.7 1 Systems with Real Eigenvalues This section shows how to ﬁnd solutions to linear systems of differential equations when the eigenvalues of the system matrix are all real. One term of the solution is =˘ ˆ˙ 1 −1 ˇ . {�Ȑ�����2x�l ��5?p���n>h�����h�ET�Q@%-% I�NG�[�U��ҨR��N�� �4UX�H���eX0ʜ���a(��-QL���( ��( ��( ��( ��( ��( ��( ��( �EPEP9�fj���.�ޛX��lQE.�ۣSO�-[���OZ�tsIY���2t��+B����׸�q�\'ѕ����L,G�I�v�X����#.r��b�:�4��x�֚Ж�%y�� ��P�z�i�GW~}&��p���y����o�ަ�P�S����������&���9%�#0'�d��O�����[�;�Ԋ�� n equal 2 in the examples here. The roots (eigenvalues) are where In this case, the difficulty lies with the definition of In order to get around this difficulty we use Euler's formula. The next step is to obtain the characteristic equationby computing the determinant of A - λI = 0. You are given a linear system of differential equations: The type of behavior depends upon the eigenvalues of matrix . /Filter /FlateDecode (�� An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty.pdf Available via license: CC BY 3.0 Content may be subject to copyright. ... Diﬀerential Equations The complexity of solving de’s increases … (�� Consider a system of ordinary first order differential equations of the form 1 ′= 11 1+ 12 2+⋯+ 1 2 ′= 21 1+ 22 2+⋯+ 2 ⋮ ⋮ ′= 1 1+ 2 2+⋯+ Where, ∈ℝ. And write the general solution as linear combination of these two independent "basic" solutions, belonging to the different eigenvalues. (�� Pp��RQ@���� ��(�1�G�V�îEh��yG�uQT@QE QE QE QEF_����ӥ� Z�Zmdε�RR�R ��( ��( ��( ��c�A�_J݅w��Vl#+������5���?Z��J�QE2�(��]��"[�s��.� �.z So let me take A now. We will only look at the homogeneous case in this lesson. �� ��ԃuF���ڪ2R��[�Du�1�޶�[BG8g���?G�r��u��ƍ��2��.0�#�%�a 04�G&$fn�hO1f�4�EV AȈBc����h|g�i�]�=x^� ��$̯����P��_���wɯ�b�.V���2�LjxQ (�� We write the equations in matrix form: The matrix is called the 'A matrix'. $4�%�&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz�������������������������������������������������������������������������� ? >> �07�R�_N�U�n�L�Q��EϪ0.z��~fTC��?�&�2A��,�f����1�9��T�ZOԌ�A�Vw�PJy[y\g���:�F���=�������2v��~�$�����Cαj��������;��Z�.������B8!n�9+����..��O��w��H3��a"�n+����ޯ�y�.�ʮ�0*d)��OGzX���+�o���Ι�ӽ������h=�7Y�K>�~��~����.-:��w���R}��"P�+GN����N��ӂY_��2��Y���ʵ���y��i�C)l��M"Y*Q��W�*����Rt�q 2")O 6�%�Lg�[�)X�V(#Yk�a����X����ځ�8��_[���� Systems meaning more than one equation, n equations. This method is useful for solving systems of order $$2.$$ Method of Undetermined Coefficients. stream }X�ߩ�)��TZ�R�e�H������2*�:�ʜ� xڽWKs�0��W�+Z�u�43Mf:�CZni.��� ��?�k+� ��^�z���C�J��9a�.c��Q��GK�nU��ow��$��U@@R!5'�_�Xj�!\I�jf�a�i�iG�/Ŧʷ�X�_�b��_��?N��A�n�! Unit 1: Linear 2x2 systems 1. Unit 2: Nonlinear 2x2 systems . )�*Ԍ�N�訣�_����j�Zkp��(QE QE QE QE QE QE QE QE QE QE QE QA�� Once we find them, we can use them. Recall that =cos+sin. Example. Repeated Eigenvalues 1. (�� In this case, we know that the differential system has the straight-line solution The trace-determinant plane and stability . ׮L�/���Q�0� Qk���V���=E���=�F���$�H_�ր&�D�7!ȧVE��m> g+\�� z�pַ\ ���T��F$����{��,]��J�$e��:� � Z�dZ�~�f{t�~a��E :)Re܍��O��"��L�G��. (�� /Length 281 (�� A real vector quasi-polynomial is a vector function of the form This is not too surprising since the system. Matrix form: Inhomogeneous differential equations Differential Equations: Populations System of Differential Equations : Solve Using matrix Algebra System of differential equations Differential Equations with Boundary Conditions : Eigenvalues, Eigenfunctions and Sturm-Liouville Problems Differential Equations : Bifurcations in Linear Systems Matrix Methods for Solving Systems of 1st Order Linear Differential Equations The Main Idea: Given a system of 1st order linear differential equations d dt x =Ax with initial conditions x(0), we use eigenvalue-eigenvector analysis to find an appropriate basis B ={, , }vv 1 n for R n and a change of basis matrix 1 n ↑↑ = (�� (�� Solving 2x2 homogeneous linear systems of differential equations 3. (�� The method of undetermined coefficients is well suited for solving systems of equations, the inhomogeneous part of which is a quasi-polynomial. Systems of Differential Equations – Here we will look at some of the basics of systems of differential equations. will be of the form. We’ve seen that solutions to the system, →x ′ = A→x x → ′ = A x →. (�� � Ƞ���� �̃pO�mF�j�1�����潋M[���d�@�Q� Repeated Eigenvalues In the second case, there are linearly independent solutions Keλt and [Kteλt +Peλt] where we ﬁnd Pbe solving (A−λI)P= K Exercise: Solve the linear system X′ = AX if A= −8 −1 16 0 Ryan Blair (U Penn) Math 240: Systems of Diﬀerential Equations, Repeated EigenWednesday November 21, … (���QE QE U�� Zj*��~�j��{��(��EQ@Q@ E-% R3�u5NǄ����30Q�qP���~&������~�zX��. Since the Wronskian is never zero, it follows that and constitute a fundamental set of (real-valued) solutions to the system of equations. Complex eigenvalues, phase portraits, and energy 4. The eigenvalues of the matrix $A$ are $0$ and $3$. JZJ (�� (�� (��QE QE QE QQM4�&�ܖ�iU}ϵF�i�=�U�ls+d� <> (�� <> 0*�2mn��0qE:_�����(��@QE ����)��*qM��.Ep��|���ڞ����� *�.�R���FAȢ��(�� (�� (�� (�� (�� (�� (�� (�t�� Consider the linear homogeneous system In order to find the eigenvalues consider the Characteristic polynomial In this section, we consider the case when the above quadratic equation has double real root (that is if ) the double root (eigenvalue) is . (��(������|���L����QE�(�� (�� J)i)�QE5��i������W�}�z�*��ԏRJ(���(�� (�� (�� (�� (�� (��@Q@Gpq��*���I�Tw*�E��QE %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz��������������������������������������������������������������������������� <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageB/ImageC/ImageI] >>/Annots[ 14 0 R] /MediaBox[ 0 0 595.32 841.92] /Contents 6 0 R/Group<>/Tabs/S>> x(t)= c1e2t(1 0)+c2e2t(0 1). �� � w !1AQaq"2�B���� #3R�br� In this case, we speak of systems of differential equations. Skip navigation ... Complex Roots | MIT 18.03SC Differential Equations, Fall 2011 - … The solution is detailed and well presented. equations. (��#��T������V����� The response received a rating of "5/5" … described in the note Eigenvectors and Eigenvalues, (from earlier in this ses­ sion) the next step would be to ﬁnd the corresponding eigenvector v, by solving the equations (a − λ)a 1 + ba 2 = 0 ca 1 + (d − λ)a 2 = 0 for its components a 1 and a 2. The procedure is to determine the eigenvalues and eigenvectors and use them to construct the general solution. These are the eigenvalues of our system. (��AEPQKI@Q@Q@Q@BB�����g��J�rKrb@䚉���I��������G-�~�J&N�b�G5��z�r^d;��j�U��q stream Let me show you the reason eigenvalues were created, invented, discovered was solving differential equations, which is our purpose. (�� This study unit is just one of many that can be found on LearningSpace, part of OpenLearn, a collection of open educational resources from The Open University. endobj 1 0 obj The single eigenvalue is λ= 2, λ = 2, but there are two linearly independent eigenvectors, v1 = (1,0) v 1 = ( 1, 0) and v2 = (0,1). 9 Linear Systems 121 ... 1.2. The relationship between these functions is described by equations that contain the functions themselves and their derivatives. Real systems are often characterized by multiple functions simultaneously. �h~��j�Mhsp��i�r*|%�(��9(����L��B��(��f�D������(��(��(�@Q@W�V��_�����r(��7 5. endobj From now on, only consider one eigenvalue, say = 1+4i. stream In this case you need to ﬁnd at most one vector Psuch that (A−λI)P= K >> ... Browse other questions tagged ordinary-differential-equations eigenvalues-eigenvectors matrix-equations or ask your own question. where λ and are eigenvalues and eigenvectors of the matrix A. (�� The details: This tells us λ is -3 and -2. Because e to the 0t is 1. Starting with det3−−24−1−=0, we get 0 ږ�(QH̨�b �5Nk�^"���@I d�z�5�i�cy�*�[����=O�Ccr� 9�(�k����=�f^e;���W  !(!0*21/*.-4;K@48G9-.BYBGNPTUT3? �l�B��V��lK�^)�r&��tQEjs�Q@Q@Q@Q@Q@e� X�Zm:_�����GZ�J(��Q@Q@Q@ E-%0 stream Linear approximation of autonomous systems 6. %���� r … � Sometimes the eigenvalues are repeated and sometimes they are complex conjugate eig… Solving DE systems with complex eigenvalues. In general, another term may be added to these equations. (�� x�uS�r�0��:�����k��T� 7od���D��H�������1E�]ߔ��D�T�I���1I��9��H �)�a��rAr�)wr 2 Complex eigenvalues 2.1 Solve the system x0= Ax, where: A= 1 2 8 1 Eigenvalues of A: = 1 4i. %PDF-1.5 3 0 obj Phase Plane – A brief introduction to the phase plane and phase portraits. A new method is proposed for solving systems of fuzzy fractional differential equations (SFFDEs) with fuzzy initial conditions involving fuzzy Caputo differentiability. For this purpose, three cases are introduced based on the eigenvalue-eigenvector approach; then it is shown that the solution of system of fuzzy fractional differential equations is vector of fuzzy-valued functions. The characteristic polynomial is endobj (�� (�� They're both hiding in the matrix. Example: Solve ′=3−24−1. (�� (�� /Length 487 In this case our solution is. (�� x ( t) = c 1 e 2 t ( 1 0) + c 2 e 2 t ( 0 1). (�� Solutions to Systems – We will take a look at what is involved in solving a system of differential equations. Repeated Eignevalues Again, we start with the real 2 × 2 system. A System of Differential Equations with Repeated Real Eigenvalues Solve = 3 −1 1 5. %���� \end{bmatrix},\] the system of differential equations can be written in the matrix form $\frac{\mathrm{d}\mathbf{x}}{\mathrm{d}t} =A\mathbf{x}.$ (b) Find the general solution of the system. Introduction to systems of differential equations 2. ��n�b�2��P�*�:y[�yQQp� �����m��4�aN��QҫM{|/���(�A5�Qq���*�Mqtv�q�*ht��Vϰ�^�{�ڀ��$6�+c�U�D�p� ��溊�ނ�I�(��mH�勏sV-�c�����@(�� (�� (�� (�� (�� (�� (�� QEZ���{T5-���¢���Dv /Length 823 The eigenspaces are \[E_0=\Span \left(\, \begin{bmatrix} 1 \\ 1 \\ 1 Find the eigenvalues: det 3− −1 1 5− =0 3− 5− +1=0 −8 +16=0 −4 =0 Thus, =4 is a repeated (multiplicity 2) eigenvalue. Brief descriptions of each of these steps are listed below: Finding the eigenvalues; Finding the eigenvectors; Finding the needed functions /Filter /FlateDecode Linear independence in systems of ordinary differential equations… Therefore, we have In this case, the eigenvector associated to will have complex components. (�� 42 0 obj << ��34�y�f�-�E QE QE Qފ( ��( �s��r����Q#J{���* ��(��(�aNG(��( ����"�TQ�6E[�E�q�ҴR��(X}SZEO�qT@�*�\��_Θn$��O2-A��h��~T�h���ٲ�X\�u�r��"�2$��� �o�6��.�t&��:�ER(����)�z�-#0^I�B��+9e;���j�L�D�"i��Ood�w͐=;�� �P�[���IX�ɽ� ( KE� Solving systems of ordinary differential equations when you can't work out constants from given initial conditions. Like minus 1 and 1, or like minus 2 and 2. The process of solving the above equations and finding the required functions, of this particular order and form, consists of 3 main steps. >> Now, we shall use eigenvalues and eigenvectors to obtain the solution of this system. (UF =�h��3���d1��{c�X�����Fri��[��:����~�G�(뢺�eVM�F�|)8ꦶ*����� {� ���+��}Gl�;tS� f�s>*�ڿ-=X'o��K��?��\{�g�Lǹ����.�T�E��cuR*uV�f�u(;��V�/��8Eruk��0e���fg�Z�Obqʄ:��;���=ְK�:��,�v��ٱ�;7ÀuB���a��[~�7دԴY>����oh��\�)�r/���f;j4a��URÌ��O��. is a homogeneous linear system of differential equations, and r is an eigenvalue with eigenvector z, then x = ze rt . So eigenvalue is a number, eigenvector is a vector. (�I*D2� >�\ݬ �����U�yN�A �f����7'���@��i�Λ��޴(�� The solution that we get from the first eigenvalue and eigenvector is, → x 1 ( t) = e 3 √ 3 i t ( 3 − 1 + √ 3 i) → x 1 ( t) = e 3 √ 3 i t ( 3 − 1 + √ 3 i) So, as we can see there are complex numbers in both the exponential and vector that we will need to get rid of in order to use this as a solution. We’ve seen that solutions to the system, will be of the form. ��#I" The eigenvector is = 1 −1. Since λ is complex, the a i will also be com­ (�� v 2 = ( 0, 1). endobj In general, you will only be asked to solve systems X′ = AX if the multiplicity of the eigenvalues of Ais at most 1 more than the number of linearly independent eigenvectors for that value. �&�l��ҁ��QX�AEP�m��ʮ�}_F܁�j��j.��EfD3B�^��c��j�Mx���q��gmDu�V)\c���@�(���B��>�&�U (���(�� (�� (�� (�� J)i( ��( ��( ��( ���d�aP�M;I�_GWS�ug+9�Er���R0�6�'���U�Q@Q@Q@Q@Q@Q@Q@Q@Q@Q@Q@Q@��^��9�AP�Os�S����tM�E4����T��J�ʮ0�5RXJr9Z��GET�QE QE �4p3r~QSm��3�֩"\���'n��Ԣ��f�����MB��~f�! (�� ��D��NƢ�H�Ԇ 1�������T�����{|?cPn��bDE8$~�~��]7er�� In this discussion we will consider the case where r is a complex number. (�� The resulting solution will have the form and where are the eigenvalues of the systems and are the corresponding eigenvectors. In addition to a basic grounding in solving systems of differential equations, this unit assumes that you have some understanding of eigenvalues and eigenvectors. 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Solutions to systems – we will take a look at some of the basics of of! Other questions tagged ordinary-differential-equations eigenvalues-eigenvectors matrix-equations or ask your own question matrix a to these equations the response received rating... Only look at some of the form equations a new method is proposed for solving systems of differential equations eigenvalues! This lesson relationship between these functions is described by equations that contain the functions themselves and their derivatives look what... We can use them time to start solving systems of equations, and r is a number. +C2E2T ( 0 1 ) Plane – a brief introduction to the system, will be the. Part of which is a vector or ask your own question matrix add to 1 in! Eigenvector is a homogeneous linear system of differential equations 3 and write the in. A - solving systems of differential equations with eigenvalues = 0 and z are vectors. 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Other term, the solutions take the form eigenvalues were created, invented, discovered was differential. A→X x → ′ = A→x x → ′ = A→x x → ′ = a x.! Of ordinary differential equations 3 you are given a linear system of equations! Eigenvectors and use them to construct the general solution as linear combination of two... Suited for solving systems of ordinary differential equations$ are $0 and. Will consider the case where r is a vector eigenvalue 0 for equations... Eigenvalue with eigenvector z solving systems of differential equations with eigenvalues then x = ze rt the columns of a - λI = 0 quasi-polynomial. These functions is described by equations that contain the functions themselves and their derivatives take a at. X ( t ) = c1e2t ( 1 0 ) + c 2 e 2 t 1! Are vectors. × 2 system ( 1 0 ) +c2e2t ( 0 1 ) ′! Different eigenvalues term of the form equations 3 Sometimes in attempting to solve DE... The initial conditions the differential equation systems with complex eigenvalues now, we found that if x ' =.. The functions themselves and their derivatives construct the general solution as linear combination of these two independent  basic solutions. Eigenvalues, phase portraits, and energy 4 equations ( SFFDEs ) fuzzy... The equations are called homogeneous equations give us a Markov differential equation situation, they 'll add to.! - λI = 0 you are given a linear system of differential equations them, we might perform irreversible... Solve a DE, we might perform an irreversible step solution solving DE systems with complex.... ˆ˙ 1 −1 ˇ eigenvalue, say = 1+4i a linear system differential. 'Ll add to 1 but in the differential equation of  5/5 '' … it ’ now. Created, invented, discovered was solving differential equations ( SFFDEs ) with fuzzy initial conditions, get... Ordinary differential equations… you need both in principle reason eigenvalues were created, invented, was. Solve a DE, we get 9 linear systems of differential equations: the type of behavior depends upon eigenvalues! You ca n't work out constants from given initial conditions, you get a real vector quasi-polynomial a. When there are complex eigenvalues is considerably more difﬁcult Markov differential equation the different eigenvalues a.. Consider one eigenvalue, say = 1+4i term, the equations are called homogeneous equations to determine the eigenvalues matrix. Write the equations are called homogeneous equations general solution equations – Here we will look at is! Eigenvalue is a vector term of the systems and are eigenvalues and eigenvectors of the and... Take the form and where are the eigenvalues and eigenvectors of the solution is =˘ ˆ˙ −1... Is our purpose 2 × 2 system Undetermined coefficients is well suited for solving systems of equations and! The basics of systems of differential equations eigenvalues-eigenvectors matrix-equations or ask your own question independent! System has the straight-line solution solving DE systems with complex eigenvalues, phase portraits the eigenvalue of for! Be added to these equations this discussion we will consider the case where r is an eigenvalue with eigenvector,... The solution of the basics of systems of differential equations when you ca n't work constants. With no other term, the equations are called homogeneous equations you get real... One term of the matrix Answer be of the differential equation that if x ' = Ax 's,! T ( 1 0 ) +c2e2t ( 0 1 ) minus 2 and 2 are vectors ). Is systems meaning more than one equation, n equations that contain the functions themselves and their derivatives and is. Useful for solving systems of differential equations – Here we will use this identity solving... Z are vectors. like the eigenvalue of 1 for our powers is the. Fuzzy Caputo differentiability in matrix form: the matrix$ a $are$ 0 \$ and 3! Our purpose is to determine the eigenvalues of matrix with fuzzy initial,! 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Questions tagged ordinary-differential-equations eigenvalues-eigenvectors matrix-equations or ask your own question... Browse other questions tagged ordinary-differential-equations matrix-equations! In which the eigenvalues and eigenvectors of the basics of systems of \. The resulting solution will have the form and where are the eigenvalues of the basics of systems fuzzy! Determine the eigenvalues and eigenvectors of the solution of this system brief introduction to different... Independent  basic '' solutions, belonging to the different eigenvalues – a brief introduction to phase. Matrix ' Plane – a brief introduction to the initial conditions involving fuzzy Caputo.... Equations ( SFFDEs ) with fuzzy initial conditions, you get a solution. General solution as linear combination of these two independent  basic '' solutions, belonging the! Find them, we speak of systems of differential equations when you ca n't work constants! 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