UICM002 & Engineering Mathematics II Unit I Ordinary Differential .
UICM002 & Engineering Mathematics - IISRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY(AN AUTONOMOUS INSTITUTION)COIMBATORE- 641010UICM002 & Engineering Mathematics – IIUnit I – Ordinary Differential EquationsCourse MaterialDifferential Equations are very important in engineering mathematics. Adifferential equations is a mathematical equation for an unknown function of one orseveral variables that relates the values of the function itself and of its derivatives ofvarious orders. It provides the medium for the interaction between Mathematics andvarious branches of science and engineering. Most common differential equations areradioactive decay, chemical reactions, Newton’s law of cooling, series RL, RC and RLCcircuits, simple harmonic motion etc.Differential EquationsAn equation which involves its derivative in it is called differential equations.There are two types they are ordinary differential equations and partial differentialequations.Ordinary Differential EquationsDifferential equations which involves only one independent variable are calledordinary differential equations.Partial Differential EquationsDifferential equations which involves more than one independent variables arecalled partial differential equations.Order of differential equationThe order of the highest differential coefficient that occurs in a differentialequation ia called order of the differential equations.Ordinary Differential Equations1
UICM002 & Engineering Mathematics - IIDegrees of differential equationThe degree of an equation is the power of the highest differential coefficientwhich occurs in it after the equation has been made rational and integer.Higher order linear differential equations with constant coefficients( )whereare constants andis either a constant or a functions ofonly.To find the complementary function (C.F)Case I:If the roots are real and unequal, sayCase II:If the roots are real and equal, say()Case III:If the roots are imaginary, say()To find the Particular Integral (P.I)Case I:( )(Case II:( ),( ))((( ))),( )Case III:( )-( ) -( )() ( ),- Ordinary Differential Equations2
UICM002 & Engineering Mathematics - IIExample: 1Solution:Given ()The auxiliary equation is,-Solving, we getThe general solution is( )Example: )([])-Ordinary Differential Equations3
UICM002 & Engineering Mathematics - IIExample: 3Solution:Given ()The auxiliary equation is,-Solving, we getThe C.F is.,()(()(())[)((-]))[[()]*() , (Ordinary Differential Equations])-4
UICM002 & Engineering Mathematics - II[()*()](,()[)()- ][][]The general solution is( )( )[]Example: 4()Solution:The auxiliary equation is,-Solving, we get(The C.F is().)(,)(-) Ordinary Differential Equations5
UICM002 & Engineering Mathematics - II {()() ()} [()(())()][ ()[()]][({())]}The general solution is( )( )(){()Ordinary Differential Equations}6
UICM002 & Engineering Mathematics - IIExample: 5()Solution:The auxiliary equation is,(-)()Solving we get(The C.F is)()(,()-,()-,(),[], (((),))-)(( )) -,-[ ]) ( (()(,.))()(( ))()()Ordinary Differential Equations7
UICM002 & Engineering Mathematics - IIExample: 6()Solution:The auxiliary equation is,-Solving we get(The C.F is).,( ) [-] (),( ),((( )-)),,Ordinary Differential Equations-8
UICM002 & Engineering Mathematics - II( )The general solution is( )( )()Example: 7Solution:Given ()The auxiliary equation is,-Solving we getThe C.F is()((),- ()),,( ([-()()))-, (])-,[Ordinary Differential Equations ]9
UICM002 & Engineering Mathematics - II) (The general solution is( )( )Example: 8()Solution:The auxiliary equation is,-Solving we getThe C.F is,-.,-.,()(-),-, [Ordinary Differential Equations- ]10
UICM002 & Engineering Mathematics - II()The general solution is( )( ),-Example: 9()Solution:The auxiliary equation is,-Solving we getThe C.F is.,()(-),()()(()())((-)[])Ordinary Differential Equations11
UICM002 & Engineering Mathematics - II()The general solution is( )( )()Example: 10()Solution:The auxiliary equation is,-Solving we getThe C.F is.,()(-),[[(-)]*() [()*(( )) , ())]([-]]Ordinary Differential Equations12
UICM002 & Engineering Mathematics - II,()((()))(-[( )],)-The general solution is( )( )[]()Example: 11()Solution:The auxiliary equation is,-Solving we getThe C.F is,-.,-.,()(-)Ordinary Differential Equations13
UICM002 & Engineering Mathematics - II()()[(([][)], ()),-]The general solution is( )( ),-[]Method of variation of parametersExample: 1Solution:()The auxiliary equation is,-Solving we getThe C.F is,-.,-Ordinary Differential Equations14
UICM002 & Engineering Mathematics - II , - (), () ()) ([-()] () ./The general solution isOrdinary Differential Equations15
UICM002 & Engineering Mathematics - II( )( )*() ./Example: 2Solution:()The auxiliary equation is,-Solving we get,The C.F is-.,-Let , - () Ordinary Differential Equations16
UICM002 & Engineering Mathematics - II () The general solution is( )( ),-Example: 3Solution:()The auxiliary equation is,-Solving we get,The C.F is-.,-Let(() )() ()(() ),,()--Ordinary Differential Equations17
UICM002 & Engineering Mathematics - II, - (), () () ()(),-- () The general solution is( ),-,()-Example: 4Solution:()The auxiliary equation isOrdinary Differential Equations18
UICM002 & Engineering Mathematics - II,-Solving we get,The C.F is-.,-Let , - [ (])) ( [ ] *{ ()()} Ordinary Differential Equations19
UICM002 & Engineering Mathematics - II , (-) [ ()()]The general solution is( )(())Example: 5Solution:()The auxiliary equation is,Ordinary Differential Equations20
UICM002 & Engineering Mathematics - IISolving we get,The C.F is-.,-Let , - () () ()The general solution is( )(Ordinary Differential Equations)21
UICM002 & Engineering Mathematics - IICauchy’s Legendre’s linear equationsExample: 1Solution:()Put(()), ()-()()The auxiliary equation is,-Solving we get,The C.F is-.,()()[][()],,Ordinary Differential Equations-22
UICM002 & Engineering Mathematics - II, (./[][*( )[] [)-( )( )] []]* ,-* ,-* [](),(( ))(())(-)The general solution is( )( )()()()()()()()Example: 2Solution:()()PutOrdinary Differential Equations23
UICM002 & Engineering Mathematics - II()()()), (()(,())(()),(-()-()-())()The auxiliary equation is,-Solving, we get,The C.F is C.F-.,()()-,-,-,,(()()--)Ordinary Differential Equations24
UICM002 & Engineering Mathematics - IIThe general solution is( )( ),( ),()-(()-)Example: 3()()Solution:(())(), ()-()(. /( ))The auxiliary equation is,-Solving, we get(The C.F is C.F)(.,)((-))Ordinary Differential Equations25
UICM002 & Engineering Mathematics - II() .()/, (*, ([))-( )( )-( )]The general solution is( )( )(( )()[)][()]Example: 4()()Solution:(())(), ()-()()The auxiliary equation is,-Solving, we get ,(-. ),Ordinary Differential Equations26
UICM002 & Engineering Mathematics - II()(),()((( )-)),()([)]The general solution is( )( )(( )( )( ))()Example: 5()Solution:()((()))Ordinary Differential Equations27
UICM002 & Engineering Mathematics - II(), ()-()()()The auxiliary equation is(),-Solving, we get()(.,)(-),( )-The general solution is( )( )(( )())Example: 6()Solution:((())), (())Ordinary Differential Equations28
UICM002 & Engineering Mathematics - II()The auxiliary equation is,-Solving, we get(The C.F is C.F)(,)(-*,- (,((.)([),( )], (-))))(),(-)()-The general solution is( )( )(( )())Example: 7(())Solution:((())), ()(Ordinary Differential Equations)29
UICM002 & Engineering Mathematics - II()()The auxiliary equation is(),-Solving, we get(The C.F is)(.,)(-)(), [(( )-])The general solution is( )( )(( )())Ordinary Differential Equations30
UICM002 & Engineering Mathematics - IIExample: 8Solution:Multiply both sides by, we get()((())), ()()Integrating, we get ( )Again Integrating, we get( )) (( )( )( )()Ordinary Differential Equations31
UICM002 & Engineering Mathematics - IIExample: 9()(),()-Solution:,()()(()(),()(,()-))()-, ()-,,-The auxiliary equation is,-Solving, we getThe C.F is.,( ),-- The general solution is( )( )( )()()Ordinary Differential Equations()()32
UICM002 & Engineering Mathematics - IIExample: 10()()()Solution:,()(()(()( ry Differential Equations33
UICM002 & Engineering Mathematics - II()()The auxiliary equation is,-Solving, we getThe C.F is.,-,en proceed,-( )(),-The general solution is( )( )(( )()()),()()-Example: 11()(),()-Solution:Ordinary Differential Equations34
UICM002 & Engineering Mathematics - II,()()(Put()(),(()-,()-))()-, ()-()()The auxiliary equation isSolving, we get,-.().,( ),-[ -] The general solution is( )( )( )()()()()Example: 12()()Ordinary Differential Equations35
UICM002 & Engineering Mathematics - IISolution:,()()(()()(,()))()-, ()-(())()The auxiliary equation isSolving, we getThe C.F is().,-,( )-,-,( )(-)(),Ordinary Differential Equations-36
UICM002 & Engineering Mathematics - II( )The general solution is( )( )()( )(())(),()- ()Example: )()()The auxiliary equation isSolving, we getThe C.F is ( )( )(.)Ordinary Differential Equations37
UICM002 & Engineering Mathematics - II((,)( )-)(()( )),-( )The general solution is( )( )( )( )()Simultaneous first order linear equations with constant coefficientsExample: 1Solution:( )( )Multiply ( ) by D, we get()( )Multiply ( ) by , we get( )( )( )( ) ( )Ordinary Differential Equations38
UICM002 & Engineering Mathematics - II()The auxiliary equation isSolving, we get,The C.F is-.,.(()()),( )-The general solution is( )( )( )()()()Using in (1), we get( )( )Example: 2Solution:( )( )Multiply ( ) by D, we get( )Ordinary Differential Equations39
UICM002 & Engineering Mathematics - II( )( )( )( )( )()The auxiliary equation isThe roots areThe C.F is.(),- (,(,),)-()(),()--,-The general solution is( )( )( )Using in (1), we get( )Example: 3()()( )()Solution:()( )Multiply ( ) by D, we get()()Ordinary Differential Equations40
UICM002 & Engineering Mathematics - II(Multiply ( ) by ()( )), we get()()()()()(( )( )( )( )(( )))(()())()((()())())( )The auxiliary equation isSolving, we get(The C.F is nary Differential Equations))))41
UICM002 & Engineering Mathematics - IIThe general solution is( )( )()()()()()( )()Adding (1) and (2), we get()()( ()()([()]))Example: 4()()Solution:()()Multiply ( ) by (( )( )), we get()()()()()( )( )( )Ordinary Differential Equations42
UICM002 & Engineering Mathematics - IIMultiply ( ) by we get()( )( )()( )( )((( )()))()()The auxiliary equation isSolving, we getThe C.F is.,( )()(([])(* )(* *([)(*( ))[-), ()( ,()))-]]Ordinary Differential Equations43
UICM002 & Engineering Mathematics - IIThe general solution is( )( )( )()From (1), we have()()( )( )(())( )Example: 5Solution:( )( )Multiply ( ) by , we get()( )( )( )( ) ( )()The auxiliary equation isOrdinary Differential Equations44
UICM002 & Engineering Mathematics - IIThe roots areThe C.F is.,-,-The general solution is( )( )( )From (1), we have( )Ordinary Differential Equations45
UICM002 & Engineering Mathematics - IITwo MarksSolution:Given ()The auxiliary equation is,-Solving, we getThe general solution is( )()()Solution:The auxiliary equation is,-Solving, we get( )( ) ( )The general solution is( ) (( ))Solution:The auxiliary equation is,-Solving, we get ()( )() ( )The general solution is( )()Ordinary Differential Equations46
UICM002 & Engineering Mathematics - II()Solution:The auxiliary equation is,By trail-is one of the root.By synthetic division method, we haveSolving, we getThe general solution is( )(())Solution:The auxiliary equation is,-Solving, we getThe general solution is( )()6. Find the particular integral of ()Solution:,(-),Ordinary Differential Equations47
UICM002 & Engineering Mathematics - II(7. Solve (,)-)Answer:The auxiliary equation is,-Solving, we get,-( )8. Find the particular integral of ()Solution:* *[) ][( )([].(]),-/Ordinary Differential Equations48
UICM002 & Engineering Mathematics - II(( )(()))()*9. Find the particular integral of (() () ion:()Ordinary Differential Equations49
UICM002 & Engineering Mathematics - II,- [] ()Solution:(),[- ] ()Ordinary Differential Equations50
UICM002 & Engineering Mathematics - II12. Reduce the equation ()into an ordinary differentialequation with constant coefficients.Answer:()(), ()-()()13. Transform the equationinto a linear equation with constantcoefficients.Answer:()(()), ()(())Answer:()(),()()(Ordinary Differential Equations)51
UICM002 & Engineering Mathematics - II()to a differential equation with constant coefficients.Answer:()(), ()-()(())()differential equation with constants coefficients.(Or)17. Transform the equation ()()into adifferential equation with constants )())18.Solution:()Ordinary Differential Equations52
UICM002 & Engineering Mathematics - IIPut(()), ()-()()The auxiliary equation is,-Solving we getThe C.F is().()Solution:()Put(()), ()-()()The auxiliary equation is,-Solving we get( )Solution:()PutOrdinary Differential Equations53
UICM002 & Engineering Mathematics - II(()), (())Integrating, we get,-Again integrating, we get( ) ( )21. Find the particular integral of (Find the particular integral of ()(Or))Solution:(,)(-) [] ()22. Find the particular integral of ()Solution:Ordinary Differential Equations54
UICM002 & Engineering Mathematics - II,()(-),23. Find the particular integral of (-)Solution:,()(-),24. Find the particular integral of (-)Solution:(,)()[ - ] Ordinary Differential Equations55
UICM002 & Engineering Mathematics - II25. Find the particular integral of ()Solution:()(,)(-) [ ( (]))APPLICATIONS OF DIFFERENTIAL EQUATIONS:The world around us is governed by differential equation .So any scientific computing willgenerally rely on a differential equation and its numerical solution in particular electrical circuits inelectrical engineering, software development in computer science engineering and various areas ofAero space Engineering, bio engineering, Chemical engineering, Industrial engineering, Mechanicaland civil Engineering. In general, modelling variations of a physical quantity, such as temperature,pressure, displacement, velocity, stress, strain, or concentration of a pollutant, with the change of timet or location, such as the coordinates (), or both would require differential equations. Similarly,studying the variation of a physical quantity on other physical quantities would lead to differentialequations. For example, the change of strain on stress for some viscoelastic materials follows adifferential equation. It is important for engineers to be able to model physical problems usingmathematical equations, andthen solve these equations so that the behaviour of the systems concerned can be studied. In thissection, a few examples are presented to illustrate how practical problems are modelledmathematically and how differential equations arise in them.Mistakes Are Proof That You Are TryingPrepared byM. Vijaya Kumar, AP / M & H / SRIT; viji778@gmail.com; vijayakumarm.sh@srit.orgOrdinary Differential Equations56
Ordinary Differential Equations 56 25. Find the particular integral of ( ) Solution: ( ) ( ) , - ( ) [ ] () APPLICATIONS OF DIFFERENTIAL EQUATIONS: . It is important for engineers to be able to model physical problems using mathematical equations, and then solve these equations so that the behaviour of the systems concerned .
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