first order linear differential equation pdf

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/FormType 1 /Name/Im1 Materials include course notes, lecture video clips, JavaScript Mathlets, practice problems with solutions, problem solving videos, and problem sets with solutions. >> 25 0 obj
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(��c�ɷ matrix-vector equation. /BaseFont/WXYYYE+CMSY10 652.8 598 0 0 757.6 622.8 552.8 507.9 433.7 395.4 427.7 483.1 456.3 346.1 563.7 571.2 812.5 875 562.5 1018.5 1143.5 875 312.5 562.5] << << /Subtype/Type1 /FirstChar 32 /BaseFont/Times-Italic /Type/Encoding /Subtype/Type1 23 0 obj /Type/Font If the diﬀerential equation does not contain (de-pend) explicitly of the independent variable or variables we call it an autonomous DE. /Widths[660.7 490.6 632.1 882.1 544.1 388.9 692.4 1062.5 1062.5 1062.5 1062.5 295.1 /Widths[272 489.6 816 489.6 816 761.6 272 380.8 380.8 489.6 761.6 272 326.4 272 489.6 x��VˎE��W�q-��|��j�+�ّ8�`-Ќ% ��'��g�i��SY�Y��Y�b��kR��/��ݏ9}�ky��6z��StmR��f�97i9��^��as9/�/?��K��(9}^,��x8N��e�*�`�wN�äw��K��V��^��C��C��D):OD΄�5ᶇh�C]PC� a��$��K�C�*Xǐa��m�7X\e��Ͳ�V;Ox�!��R�5z,��ӷ��-�� 694.5 295.1] 500 333 1000 500 333 944 250 250 250 250 333 333 556 556 350 500 889 333 980 389
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/BaseFont/FFECLP+CMBX12 << 2. 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 endobj Non-Linear, First-Order Diﬁerential Equations In this chapter, we will learn: 1. 22 0 obj <> ��!g�� il��P�c':DC�0�t=��Xv�q=�q�q6DYQ�#6mt4m��{If�P`�=K��XR� BV��s,��& �� d�fD(h�{f0 ��3�@�>�@�:wb�K�($��!dȣ�+]�3oh�X�\��똺��!w��J�.Ʀ`d'�f�~��6 ��W�zb�ߙ��졵I�D�g��S ��K�L漶��6- � �Ӏ�F�W�Gt�6j��B��J�Dt��S��$�� Ӣ3��|�|�\Id�G�%��! 295.1 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 531.3 295.1 /Filter/FlateDecode /FirstChar 33 /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 << The differential equation in the picture above is a first order linear differential equation, with $P(x) = 1$ and $Q(x) = 6x^2$. 544 516.8 380.8 386.2 380.8 544 516.8 707.2 516.8 516.8 435.2 489.6 979.2 489.6 489.6

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Equation (d) expressed in the “differential” rather than “difference” form as follows: 2 ( ) 2 2 h t D d g dt dh t ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ =− (3.13) Equation (3.13) is the 1st order differential equation for the draining of a water tank. Use of phase diagram in order to under-stand qualitative behavior of diﬁerential equation. 0 0 0 0 0 0 0 0 0 0 777.8 277.8 777.8 500 777.8 500 777.8 777.8 777.8 777.8 0 0 777.8 /Name/F4 275 1000 666.7 666.7 888.9 888.9 0 0 555.6 555.6 666.7 500 722.2 722.2 777.8 777.8 /FontDescriptor 9 0 R

We'll talk about two methods for solving these beasties. 761.6 489.6 516.9 734 743.9 700.5 813 724.8 633.9 772.4 811.3 431.9 541.2 833 666.2 16 0 obj :�j` endobj endobj 389 389 278 500 444 667 444 444 389 400 275 400 541 250 250 250 333 500 556 889 500 endobj /FontDescriptor 12 0 R >> 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 << stream endobj /FontDescriptor 18 0 R AM]]��e6��䤬黅(��KS�L��Ыm-�@��\��~�? /Name/R4 19 0 obj 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 642.9 885.4 806.2 736.8 380.8 380.8 380.8 979.2 979.2 410.9 514 416.3 421.4 508.8 453.8 482.6 468.9 563.7 492.9 510.4 505.6 612.3 361.7 429.7 553.2 317.1 939.8 644.7 513.5 534.8 474.4 479.5 /Length 953 %PDF-1.3 stream >> 444.4 611.1 777.8 777.8 777.8 777.8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

�>߰��d��y22��$�%ܓg�lᵇ�鵓��T�_�f�Ք�3\��{�xþ-��eޏ�B�!�6K) /Encoding 7 0 R /FirstChar 33 << 699.9 556.4 477.4 454.9 312.5 377.9 623.4 489.6 272 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 How to solve nonlinear ﬂrst-order dif-ferential equation? Convert the third order linear equation below into a system of 3 first order equation using (a) the usual substitutions, and (b) substitutions in the reverse order: x 1 = y″, x 2 = y′, x 3 = y. Deduce the fact that there are multiple ways to rewrite each n-th order linear equation into a linear system of n equations. Otherwise, we are dealing with a non-homogeneous linear DE. >> /Type/Font endobj /Subtype/Form

0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 777.8 777.8 1000 500 500 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 << As a consequence, the DE (1.2), is non-autonomous. x��\Ys��3+?b�ݔv�Uy�� x$b�� R"�*W�k��__��U� ��A��;;x{�;��ћ��ͭ��vuxz�w�[�E��j'�j�+��N��ᛃ��z��Y&hg�q�ǃ/�9�¯��n�[)ՙ��0l�JikVzދ��띖��g��O�}��'�I!��z3��c«� ��F�q�W�~��7W믦e�d��v��ܥak�\o��_o6��^K�>�L�'�"�l��˼��xO��AO;'�Wy��49SY /Font 25 0 R /Matrix[1 0 0 1 -239 -458] /LastChar 196 /Type/Font

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