Second Order Differential Equation Non Homogeneous

1 Class Notes 5: Second Order Differential Equation Non Homogeneous 82A Engineering Mathematics Second Order Linear Differential equations Homogeneous & Non Homogenous v p, q, g are given, continuous functions on the open interval I )(0)()(tgytqytpyHomogeneousNon-homogeneo us Solution:where yc(x): solution of the Homogeneous Equation (complementary solution)yp(x): anysolution of the non- Homogeneous Equation (particular solution) sHomogeneoushomogeneou-Non , 0 , )()()(xgyxqyxpy)()(xyxyypc Second Order Linear Differential equations Homogeneous & Non Homogenous Structure of the General Solution 00)0()0( ytyyty Order Linear Differential equations Non Homogenous )()()(tftqytpy 00)0()0( ytyyty ( ) If Y1and Y2are solutions of the nonhomogeneous Equation Then Y1 -Y2is a solution of the Homogeneous Equation If, in addition, {y1, y2} forms a fundamental solution set of the Homogeneous Equation , then there exist constants c1and c2 such that)()()()(221121tyctyctYtY )()()(tgytqytpy 0)()( ytqytpyTheorem ( )

2 General Solution The general solution of the nonhomogeneousequation can be written in the formwhere y1and y2form a fundamental solution set for the Homogeneous Equation , c1and c2 are arbitrary constants, and Y(t) is a specific solution to the nonhomogeneous Equation .)()()()(2211tYtyctycty )()()(tgytqytpy The methods of undetermined coefficients The methods of variation of parameters Second Order Linear Non Homogenous Differential equations Methods for Finding the Particular Solution Make an initial assumption about the format of the particular solution Y(t) but with coefficients left unspecifiedSubstitute Y(t) into y + p(t)y + q(t)y = g(t) and determine the coefficients to satisfy the equationThere is no solution of the form that we assumedFind a solution of Y(t)Determine the coefficientsEndNYSecond Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Block Diagram Advantages Straight Forward Approach -It is a straight forwardto execute once the assumption is made regarding the form of the particular solution Y(t)

3 Disadvantages Constant Coefficients - Homogeneous equations with constant coefficients Specific Nonhomogeneous Terms-Useful primarily for 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 The particular solution ypfor the nonhomogeneous Equation Class A)(xgcyybya )()(110 xin Polynomial 0 ..0,0 ..0 ..11202110110bcAxAxAxbcAxAxAxcAxAxAynnnn nnnnpSecond Order Linear Non Homogenous Differential equations Particular Solution For Non Homogeneous EquationClass A The particular solution ypfor the nonhomogeneous Equation Class B ).

4 ()()(110nnnxnxaxaxaexPexg 0)( )..()(110 chAxAxAexgnnnxequationstic characteri the of root a not is 0)( )..()(110xchAxAxAxexgnnnxequationstic characteri the of root simple a is 0)( )..()(1102xchAxAxAexxgnnnxequationstic characteri the of root double a is Second Order Linear Non Homogenous Differential equations Particular Solution For Non Homogeneous EquationClass B)(xgcyybya The particular solution ypfor the nonhomogeneous Equation Class C )..()(cos sin)(110nnnxinxaxaxaexPxxexg or 0)( ;.. )( ;.. ichBxBxBxAxAxAxxeichBxBxBxAxAxAxeynnnnnn xnnnnnnxp)(xgcyybya Second Order Linear Non Homogenous Differential equations Particular Solution For Non Homogeneous EquationClass C The particular solution ofs is the smallest non-negative integer (s=0, 1, or 2) that will ensure that no term in Yi(t)is a solution of the corresponding Homogeneous equations is the number of time0 is the root of the characteristic Equation is the root of the characteristic Equation +i is the root of the characteristic Equation )(tgcyybyai )(tgi)(tYinnnnatatatP.)

5 (110 nnnsAtAtAt ..110tnetP )( tnnnseAtAtAt ..110 ttetPtn cossin)( teAtAtAttnnns teBtBtBtnnn Second Order Linear Non Homogenous Differential equations Particular Solution For Non Homogeneous EquationSummarySecond Order Linear Non Homogenous Differential equations Particular Solution For Non Homogeneous EquationExamplesteyyy2343 0432 25232)4(493 Second Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 1teyyy2343 tttAetYAetYAetY2224)(2)()( ttAeeAAA2263464 21 AtpetY221)( Second Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 1 tAtYtAtYtAtYsin)(cos)(sin)(0cos3sin)52(s in2sin4cos3sin tAtAttAtAtAAssumeThere is no choice for constant A that makes the Equation true for all tSecond Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 2tyyysin243 Assume tBtAtYtBtAtYtBtAtYcossin)(sincos)(cossin )( Second Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 2tyyysin243 tttYpcos173sin175)( 053235 BABAttBABtABAsin2cos)43(sin)43( 173175 BAteyyyt2cos843 01028210 BABAS econd Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 3 teBAteBAtYteBAteBAtYtBetAetYtttttt2sin)3 4(2cos)43()(2sin)2(2cos)2()(2sin2cos)(13 2.)

6 1310 BAtetetYttp2sin1322cos1310)( Second Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 4 (Pathological Case) Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 4 (Pathological Case) Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 5(Pathological Case) ZillSecond Order Linear Non Homogenous Differential equations Method of Undermined Coefficients Example 6 (Pathological Case) Zillg(x)Advantage General methodDiff. eq.)()()(tgytqytpy For the Homogeneous diff. )()( ytqytpythe general solution is)()()(2211tyctyctyc 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 constant by function21&cc)(),(21tutu)()(2211tuctuc )()()()((*)2211tytutytuyp -Find such that is the solution to the nonhomogeneous diff.

7 Than the Homogeneous eq.)(),(21tutu222222221111111122221111yu yuyuyuyuyuyuyuyyuyuyuyuypp Second Order Linear Non Homogenous Differential equations Method of Variation of Parameters 2211221122221111222211112211222211112222 222211111111])[()()()(yuyuyuyupyuyuyuyuq yypyuqyypyuyuyuxqyuyuyuyuxpyuyuyuyuyuyuy uyuyxqyxpyppp -Seek to determine 2 unknown function -Impose a condition-The two Eqs. )(221122112211tgyuyuyuyupyudxdyudxd = 0= 0 )(),(21tutu0)()()()(2211 tytutytu )()()()()(0)()()()(22112211tgtytutytutyt utytu212121,,,,,uuyyyy knownunknownSecond Order Linear Non Homogenous Differential equations Method of Variation of Parameters-Seek to determine 2 unknown function -Impose a condition Reducing the diff. Equation to-The two Eqs. )(221122112211tgyuyuyuyupyudxdyudxd )(),(21tutu0)()()()(2211 tytutytu )()()()()(0)()()()(22112211tgtytutytutyt utytu212121,,,,,uuyyyy knownunknown )(221122112211tgyuyuyuyupyuyudxd Second Order Linear Non Homogenous Differential equations Method of Variation of Parameters)()()()()(2211tgtytutytu 212111221212210;0yyyygyyuyyyyygyu ),(;),(21122121yyWgyuyyWgyu 2211212121),(;),(cdtyyWgyucdtyyWgyu)()() ()(2211tytutytuyp (*)onBased 2211212121),(),()(cdtyyWgyycdtyyWgyytYpS econd Order Linear Non Homogenous Differential equations Method of Variation of ParametersTheorem ( ) Consider the equations If the functions p, qand gare continuous on an open interval I, and if y1and y2are fundamental solutions to Eq.

8 (2), then a particular solution of Eq. (1) is and the general solution is dttyyWtgtytydttyyWtgtytytY)(,)()()()(,)( )()()(21122121)()()()(2211tYtyctycty )2(0)()()1()()()( ytqytpytgytqytpy 2,002121 eeeeeeyyyyeeWxxxxxxxyy1 Second Order Linear Non Homogenous Differential equations Method of Variation of Parameters Example xxCececy 211;101212 -Solution to the homogeneousdiff Eq. -Solution to the nonhomogeneousdiff Eq. dtteuxeexeuxxtxxx 0212)1(21011dtteuxexeeuxxtxxx 0212)1(21022,2121)(00dtteedtteetYyxxtxxx txp Second Order Linear Non Homogenous Differential equations Method of Variation of Parameters Example dttyyWtgtytydttyyWtgtytytYyp)(,)()()()(, )()()()(21122121)()()()()(2211tytutytutY yp dtteedtteeececyxxtxxxtxxx 00212121 Second Order Linear Non Homogenous Differential equations Method of Variation of Parameters Example pcyyy -General Solution to the nonhomogeneousdiff Eq.

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