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Class Notes 5:Second Order Differential Equation –Non Homogeneous82A – Engineering Mathematics

Second Order Linear Differential Equations –Homogeneous & Non Homogenous vHomogeneous 0y p(t ) y q(t ) y g (t ) Non-homogeneous p, q, g are given, continuous functions on the open interval I

Second Order Linear Differential Equations –Homogeneous & Non Homogenous –Structure of the General Solution g ( x) , Non - homogeneous y p ( x) y q ( x) y Homogeneou s 0, I.C. y (t 0) y0 y (t 0) y0 Solution:y yc ( x) y p ( x)whereyc(x): solution of the homogeneous equation (complementary solution)yp(x): any solution of the non-homogeneous equation (particular solution)

Second Order Linear Differential Equations –Non Homogenousy p(t ) y q(t ) f (t )I.C. y (t 0) y0 y (t 0) y0

Theorem (3.5.1) If Y1 and Y2 are solutions of the nonhomogeneous equationy p(t ) y q(t ) y g (t ) Then Y1 - Y2 is a solution of the homogeneous equationy p(t ) y q(t ) y 0 If, in addition, {y1, y2} forms a fundamental solution set of thehomogeneous equation, then there exist constants c1 and c2 suchthatY1 (t ) Y2 (t ) c1 y1 (t ) c2 y2 (t )

Theorem (3.5.2) – General Solution The general solution of the nonhomogeneous equationy p(t ) y q(t ) y g (t )can be written in the formy(t ) c1 y1 (t ) c2 y2 (t ) Y (t )where y1 and y2 form a fundamental solution set for the homogeneousequation, c1 and c2 are arbitrary constants, and Y(t) is a specificsolution to the nonhomogeneous equation.

Second Order Linear Non Homogenous Differential Equations –Methods for Finding the Particular Solution The methods of undetermined coefficients The methods of variation of parameters

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Block DiagramMake an initial assumption about the format of the particularsolution Y(t) but with coefficients left unspecifiedSubstitute Y(t) into y’’ p(t)y’ q(t)y g(t) and determine thecoefficients to satisfy the equationThere is nosolution of theform that weassumedNDeterminethecoefficientsYFind asolution ofY(t)End

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Block Diagram Advantages– Straight Forward Approach - It is a straight forward to executeonce the assumption is made regarding the form of the particularsolution Y(t) Disadvantages– Constant Coefficients - Homogeneous equations with constantcoefficients– Specific Nonhomogeneous Terms - Useful primarily forequations for which we can easily write down the correct form ofthe particular solution Y(t) in advanced for which theNonhomogenous term is restricted to Polynomic Exponential Trigonematirc (sin / cos )

Second Order Linear Non Homogenous Differential Equations –Particular Solution For Non Homogeneous EquationClass A The particular solution yp for the nonhomogeneous equationay by cy g (x) Class A Pn ( x) Polynomial in xg ( x) nn 1ax ax . an01 A0 x n A1 x n 1 . An y p x A0 x n A1 x n 1 . An x 2 A x 2 A x n 1 . A01n c 0c 0, b 0c b 0

Second Order Linear Non Homogenous Differential Equations –Particular Solution For Non Homogeneous EquationClass B The particular solution yp for the nonhomogeneous equationay by cy g (x) Class B e x Pn ( x)g ( x ) xnn 1e(ax ax . an )01 e x ( A0 x n A1 x n 1 . An )g ( x) is not a root of the characteri stic equation ch ( ) 0 xe x ( A0 x n A1 x n 1 . An )g ( x) is a simple root of the characteri stic equation ch ( x) 0 x 2 e x ( A0 x n A1 x n 1 . An )g ( x) is a double root of the characteri stic equation ch ( x) 0

Second Order Linear Non Homogenous Differential Equations –Particular Solution For Non Homogeneous EquationClass C The particular solution yp for the nonhomogeneous equationay by cy g (x) Class C e x sin x or cos x Pn ( x)g ( x) i xnn 1e(ax ax . an )01 x sin x A0 x n A1 x n 1 . An e ;nn 1 cos x B0 x B1 x . Bn yp nn 1 sin xAx Ax . An 01 xe x ;nn 1 cos x B0 x B1 x . Bn ch ( i ) 0ch ( i ) 0

Second Order Linear Non Homogenous Differential Equations –Particular Solution For Non Homogeneous EquationSummary The particular solution ofay by cy g i (t )Yi (t )g i (t )Pn (t ) a0t n a1t n 1 . anPn (t )e t t sin tPn (t )e cos t t A t A t . A et A t A t . A e cos t B t B t . B e sin t t s A0t n A1t n 1 . Anssn01nn 1 t1n tn 1n0 tn 1n01ns is the smallest non-negative integer (s 0, 1, or 2) that will ensure that no term inYi(t) is a solution of the corresponding homogeneous equations is the number of time0 is the root of the characteristic equationα is the root of the characteristic equationα iβ is the root of the characteristic equation

Second Order Linear Non Homogenous Differential Equations –Particular Solution For Non Homogeneous EquationExamples

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 1y 3 y 4 y 3e 2t 2 3 4 03 9 4( 4) 3 5 22 2

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 1y 3 y 4 y 3e 2t Y (t ) Ae 2t 2t Y (t ) 2 Ae Y (t ) 4 Ae 2t 4 A 6 A 4 A e 2t 3e 2t 6 A1A 21 2tYp (t ) e2

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 2y 3 y 4 y 2 sin tAssume Y (t ) A sin t Y (t ) A cos t Y (t ) A sin t A sin t 3 A cos t 4 A sin t 2 sin t(2 5 A) sin t 3 A cos t 0There is no choice for constant A that makes the equation true for all t

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 2y 3 y 4 y 2 sin tAssume Y (t ) A sin t B cos t Y (t ) A cos t B sin t Y (t ) A sin t B cos t ( A 3B 4 A) sin t ( B 3A 4B) cost 2 sin t 5 A 3B 2 3 A 5 B 0A 5B 31717Yp (t ) 5 sin t 3 cost1717

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 3y 3 y 4 y 8et cos 2t Y (t ) Ae t cos 2t Be t sin 2t tt Y (t ) ( A 2 B)e cos 2t ( 2 A B)e sin 2t Y (t ) ( 3 A 4 B)et cos 2t ( 4 A 3B)et sin 2t 10 A 2 B 8 2 A 10B 0A 10 13; B 2 1310 t2 tYp (t ) e cos 2t e sin 2t1313

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 4(Pathological Case) – Zill p.153

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 4(Pathological Case) – Zill p.153

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 5(Pathological Case) – Zill

Second Order Linear Non Homogenous Differential Equations –Method of Undermined Coefficients – Example 6(Pathological Case) – Zillg(x)

Second Order Linear Non Homogenous Differential Equations –Method of Variation of ParametersAdvantage – General methodDiff. eq.y p(t ) y q(t ) y g (t )For the Homogeneous diff. eq.y p(t ) y q(t ) y 0the general solution isyc (t ) c1 y1 (t ) c2 y2 (t )so far we solved it for homogeneous diff eq. with constant coefficients.(Chapter 5 – non constant – series solution)

Second Order Linear Non Homogenous Differential Equations –Method of Variation of ParametersReplace the constantc1 & c2by function u1 (t ), u2 (t )c1 u1 (t )c2 u2 (t )(*) y p u1 (t ) y1 (t ) u2 (t ) y2 (t )-Find u1 (t ), u2 (t ) such that is the solution to the nonhomogeneous diff. eq.rather than the homogeneous eq.y p u1 y1 u1 y1 u2 y2 u2 y2y p u1 y1 u1 y1 u1 y1 u1 y1 u2 y2 u2 y2 u2 y2 u2 y2

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parameters①②⑤②①⑤③③y p p ( x) y p q ( x) y p u1 y1 u1 y1 u1 y1 u1 y1 u2 y2 u 2 y2 u 2 y2 u2 y2①④①① p ( x) u1 y1 u1 y1 u2 y2 u 2 y2 ①④ q ( x)[u1 y1 u2 y2 ]①②④③⑤ u1 y1 py1 qy1 u2 y2 py2 qy2 u1 y1 u1 y1 u2 y2 u2 y2 p u1 y1 u 2 y2 u1 y1 u 2 y2 0 0d u1 y1 d u2 y2 p u1 y1 u2 y2 u1 y1 u2 y2 g (t )dxdx-Seek to determine 2 unknown function u1 (t ), u2 (t )Impose a condition u1 (t ) y1(t ) u2 (t ) y2 (t ) 0y1 , y2 , y1 , y2 ,The two Eqs. u1 (t ) y1 (t ) u2 (t ) y2 (t ) 0 u1 (t ) y1 (t ) u2 (t ) y2 (t ) g (t )u1 , u 2 knownunknown

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parametersd u1 y1 d u2 y2 p u1 y1 u2 y2 u1 y1 u2 y2 g (t )dxdxd u1 y1 u2 y2 p u1 y1 u2 y2 u1 y1 u2 y2 g (t )dx-Seek to determine 2 unknown function u1 (t ), u2 (t )-Impose a conditionu1 (t ) y1(t ) u2 (t ) y2 (t ) 0Reducing the diff. equation tou1 (t ) y1 (t ) u2 (t ) y2 (t ) g (t )-The two Eqs. u1 (t ) y1 (t ) u2 (t ) y2 (t ) 0 u1 (t ) y1 (t ) u2 (t ) y2 (t ) g (t )y1 , y2 , y1 , y2 ,u1 , u 2 knownunknown

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parameters0u1 gy2y2 y1y1 y2y2 u1 y2 g;W ( y1, y2 )u1 ;u2 u2 y2 gdt c1;W ( y1, y2 )y1y1 gy1y1 y2y2 0y1gW ( y1, y2 )u2 y1gdt c2W ( y1, y2 )Based on (*) y p u1(t ) y1(t ) u2 (t ) y2 (t )Y p (t ) y1 y2 gy1gdt c1 y2dt c2W ( y1, y2 )W ( y1, y2 )

Theorem (3.6.1) Consider the equationsy p(t ) y q(t ) y g (t )y p(t ) y q(t ) y 0(1)(2) If the functions p, q and g are continuous on an openinterval I, and if y1 and y2 are fundamental solutions to Eq.(2), then a particular solution of Eq. (1) isy2 (t ) g (t )y1 (t ) g (t )Y (t ) y1 (t ) dt y2 (t ) dtW y1 , y2 (t )W y1 , y2 (t )and the general solution isy(t ) c1 y1 (t ) c2 y2 (t ) Y (t )

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parameters – Exampley y -1xSolution to the homogeneous diff Eq. 2 1 0 1 1; 2 1yC c1e x c2e x-Solution to the nonhomogeneous diff Eq. xW e ,e x y1 y1 y2 e x xy2 ee x e x e 0 e 0 2

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parameters – Exampleu1 e x01x e x 2 ex x(1 x)1 u1 22 e tdttx0ex01xexu2 2xxe (1 x)1 u2 22 etdttx0y p Y (t ) y1 (t ) y2 (t ) g (t )y (t ) g (t )dt y2 (t ) 1dtW y1 , y2 (t )W y1 , y2 (t )y p Y (t ) u1 (t ) y1 (t ) u2 (t ) y2 (t )1 x e t1 x ety p Y (t ) e dt e dt,2 x0 t2 x0 txx

Second Order Linear Non Homogenous Differential Equations –Method of Variation of Parameters – Example-General Solution to the nonhomogeneous diff Eq.y yc y p1 x e t1 x etx xy c1e c2 e e dt e dt2 x0 t2tx0xx

equations for which we can easily write down the correct form of the particular solution Y(t) in advanced for which the Nonhomogenous term is restricted to Polynomic Exponential Trigonematirc (sin / cos ) Second Order Linear Non Homogenous Differential Equations – Method of Undermined Coefficients –Block Diagram

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