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# Ordinary Differential Equation MATH 335

GPA 3.73

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This 6 page Class Notes was uploaded by Dr. Cleo Wunsch on Thursday October 15, 2015. The Class Notes belongs to MATH 335 at New Mexico Institute of Mining and Technology taught by Staff in Fall. Since its upload, it has received 47 views. For similar materials see /class/223630/math-335-new-mexico-institute-of-mining-and-technology in Mathematics (M) at New Mexico Institute of Mining and Technology.

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Date Created: 10/15/15

Ivan Avramidi MATH 335 Applied Analysis FORMULAS l MATH 335 Applied Analysis I Formulas First Order Eqs Separable Eqs yNsdsmMudu0 Linear Eqs Substitution y 01 linear eq 3 lt17 ngtpltzgtv lt17 71W Riccati Eq 21 fz gzy M90112 lf y1 is a solution7 then substitution y yl i linear eq d1 E 996 2hyi9 li 41W Homogeneous Eqs dy y F H 7 dx x Mm yo Substitution separable eq d g F1 7 1 Orthogonal trajectories fz7y0 61115214 dz awf Exact Eqs M7yd96N7ydy 0 Ivan Avramidi MATH 335 Applied Analysis FORMULAS Check 5yM967y azNWy Solution WM O 7 where ag96711 Mz7y7 1MdzNdyi5dexdy Integrating factor 5yqziy MM um alum If My 7 IvaN may then u we d Lr M N If Nm 7 MW hy7 then M My du Nm 7 My d M M y Autonomous Eqs dyi dx 7 y Equilibrium solutions Critical points y 0 Second Order Eqs Eqs With Dependent Variable Missing 2quot f h 1 Substitution Mt y t 1 ft1 m0mwmo Eqs With Independent Variable Missing 2 f y y Substitution My y d 71 fy7v d t F 0 Second Order Linear Eqs Homogeneous Eq Ly y pty qty 0 Fundamental Solutions y1 y2 Wronskian Wy17y2 91 1 7amp0 yl y General Solution Mt Gil t 021209 Homogeneous Eqs With Constant Coeff ay by cy0 Characteristic Eq y 6 ar2 brc 0 1 r1 E fl i Vb 7 406 Fundamental Solutions l Real Distinct Roots r1 31 Q 91 5M7 12 5m Ivan Avramidi MATH 335 Applied Analysis FORMULAS 3 ll Complex Distinct Roots r12 Aim u 31 0 lyl 6M cosut7 yg 6M sinut I lll Repeated Roots r1 r2 r 91 5S7 12 t5 39rt Euler Eq tzy aty By 0 Substitution z lnt Characteristic Eq y t7 rr71ar 0 Real Distinct Roots r1 31 Q 91 717 12 1 Complex Distinct Roots r12 A i m yl t cosu ln t7 yg t sinu ln t Repeated Roots r1 r2 r y1tr y2trlnt Reduction of Order lf yl is solution of y WW W 0 then i DH p2amp10 yr Nonhomogeneous Eq 1 pty qty 9t Particular Solution Yt General Solution Mt Gil t 021209 Y 129 119 Y t 7 dt t dt ylWQhyz 120 Mal17112 g Pm Y mm 9 Pnte t gt Y tsQnte t g Pnte imt gt Y tsQnt em cos t t5RnteD t sin t Higher Order Linear Eqs Homogeneous Eq Ly W pn71ty 1 poty 0 Fundamental Solutions yl yn Wronskian Wy177yn y 0 yieil 39 39 172571 General Solution Homogeneous Eqs With Constant Coeff lanym an71yltnelgtru 00y 0 Characteristic Eq y e 1 anrnan1r a1ra00 Fundamental Solutions l Real Distinct Roots r1 31 r2 ll Complex Distinct Roots r12 Aim u 31 0 yl 6M COSMt yg 6M sinut I Ivan Avramidi MATH 335 Applied Analysis FORMULAS 4 HI Repeated Roots r with multiplicity s y1 6 22 Minus ts lequot Repeated Roots A i m with multiplicity s y1 e cosut ys tsile cosut I y51 e sinut ygs ts le sinut I Nonhomogeneous Eq Ly all pn71ty 1gt poty 9t Particular Solution Yt General Solution ylttgt c1y1lttgt mat Ylttgt Wk is obtained from W by replacing the kth col umn by 0 01 g Pm t gt g Pm t 6 gt g Pm Demim gt tsRmte t sin t Series Solutions of Linear Eqs Convention 1 0 k 12 Power series 2 MW 0 Za z 7 x0 Radius of convergence Multiplication of series 2 anx 7 0 Zbkw 7 0k Z 0z 7 0 Taylor series k0 f is analytic at 0 if Taylor series converges to f for lx 7 ml lt R Shift of summation index 2 anx 7 x0 N Z anN 7 0 2nd Order Linear Eq Fwy Qxy Rzy 0 Ordinary Point PWO 7 0 P QR analytic at 0 Clo y07 11 91 Regular Singular Point x0 0 P0 0 QW 2PM mpw and z PW analytic at 0 7 r 7 i 2 po7ilgr xpm7 10753596 Pm lndicial Equation 77 1P07 QO0 Ivan Avramidi MATH 335 Applied Analysis FORMULAS Exponents at Singularity Case Ill 71 r2 T12 g 100 12 i 100 12 i 410 T1 T2 100 12 i 4 Fundamental Solutions Case I T1773 real7 r1 7 r2 not integer General Solution 2w lnx en n0 d07 60 arbitrary General Solution Case IV 71 7 r2 N positive integer d07 60 arbitrary Case ll r1775 complex y2z ayl lnz Z cnxwm n0 6 an a0 7 r2anr 772 a lirn 7quot 7 r2aNr 7772 10001 General Solution General Solution for T1 a i 2396 N 1 co Z dnxw 2 Z en ln z fn zn v z n0 n0 Z dn cos ln s en sin ln znw n0 d07 60 arbitrary d07 60 arbitrary7 real Ivan Avramidi MATH 335 Applied Analysis FORMULAS Laplace Transform 716709F8 6t 7 cft 7 c 00 an i am 7 M 7 5 dt w W 7 rims ft Wt Cft CF5 Sinat rims cm 5 a cosat m flt Jaw F15 F25 Impulse Dirac Function 71F18 F28 f1t f2t 00 6t7t0dt1 WW 5W 11 me WW 6t 7 toft dt m0 t ft 1 Fn8 71Fn8 it ft W i to d 39 1to 6dft F5 i 0 6t 7 t0 6t0e 9 0 Convolution rims 7 cgt mm mm 7 E W 7ltmgtlttgt 7 fltt7gtglt gtd rlmks 1 t g Step Heaviside Function f9gf f9192f91f92 0 K f9hf9h uct6t7c17 D 7 7 00 am 7 cgt 7 0t 7 cft 7 0 e Fs Af 9X10 ft 9t Hs F5Gs

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