Chapter 11 Linear Differential Equations Of Second And .

Chapter 11Linear Differential Equations of Secondand Higher Order11.1 IntroductionA differential equation of the form 0 in which thedependent variableand its derivatives viz.,etc occur in firstdegree and are not multiplied together is called a Linear Differential Equation.11.2 Linear Differential Equations (LDE) with Constant CoefficientsA general linear differential equation of nth order with constant coefficients isgiven by:whereare constant andOr, wheredifferential operators.is a function ofalone or constant., .,,are called11.3 Solving Linear Differential Equations with Constant CoefficientsComplete solution of equationis given byC.F P.I.where C.F. denotes complimentary function and P.I. is particular integral.When, then solution of equationis given byC.F11.3.1 Rules for Finding Complimentary Function (C.F.)Consider the equationStep 1: Put, auxiliary equation (A.E) is given by ③Step 2: Solve the auxiliary equation given by ③Page 1

I.II.III.If the n roots of A.E. are real and distinct sayC.F. If two or more roots are equal i.e. C.F. If A.E. has a pair of imaginary roots i.e., ,,,C.F. IV.If 2 pairs of imaginary roots are equal i.e.,C.F. Example 1 Solve the differential equation:Solution:Auxiliary equation is:C.F. Sincesolution is given byC.FExample 2 Solve the differential equation:Solution: .①Auxiliary equation is:By hit and trialis a factor of ① ① May be rewritten asPage 2

C.F. Sincesolution is given byC.FExample 3 SolveSolution: Auxiliary equation is: .①is a factor of ①By hit and trial ① May be rewritten as ②is a factor of ②By hit and trial ② May be rewritten asC.F. Sincesolution is given byC.FExample 4 Solve the differential equation:Solution:Auxiliary equation is:Page 3

C.F. Sincesolution is given byC.FExample 5 Solve the differential equation:Solution:Auxiliary equation is:C.F. Sincesolution is given byC.FExample 6 Solve the differential equation:Solution: .①Auxiliary equation is:By hit and trialis a factor of ① ① May be rewritten asPage 4

C.F. Sincesolution is given byC.FExample 7 Solve the differential equation: .①Solution: Auxiliary equation is:Solving ①, we getC.F. Sincesolution is given byC.F]Example 8 Solve the differential equation: .①Solution: Auxiliary equation is:Solving ①, we getC.F. Sincesolution is given byC.F11.3.2 Shortcut Rules for Finding Particular Integral (P.I.)Consider the equationThen P.I ,, Clearly P.I. 0 ifCase I: WhenUse the rule P.I ,In case of failure i.e. ifPage 5

P.I If ,, P.I. ,and so onExample 9 Solve the differential equation:Solution:Auxiliary equation is:C.F. P.I. , by puttingComplete solution is:C.F. P.I Example 10 Solve the differential equation:Solution:Auxiliary equation is:C.F. P.I. , putting P.I P.I. ,if ,P.I. Page 6

Complete solution is: C.F. P.I Example 11 Solve the differential equation:Solution:Auxiliary equation is:C.F. P.I. ( and Puttingandandin the three terms respectivelyfor first two terms P.I.ifNow puttingP.I.in first two terms respectively P.I.P.I.Page 7

Complete solution is:C.F. P.I Case II: WhenIf, put,, ,This will form a linear expression in in the denominator. Now rationalize thedenominator to substitute. Operate on the numerator term by termby takingIn case of failure i.e. ifP.I.,If, P.I.,Example 12 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. puttingP.I , Rationalizing the denominator, Putting) )Page 8

Complete solution is:C.F. P.I )Example 13 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Puttingin the 1st term andin the 2nd termP.I , Rationalizing the denominator Now, PuttingC.F. P.IExample 14 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Page 9

Puttingin the 1st term andin the 2nd termfor the 1st termWe see thatP.I.P.I.,P.I. Complete solution is:C.F. P.ICase III: WhenP.I 1. Take the lowest degree term common fromto get an expressionof the formin the denominator and take it to numerator tobecome2. Expandusing binomial theorem up to nth degree as(n 1)th derivative ofis zero3. Operate on the numerator term by term by takingFollowing expansions will be useful to expandpowers ofin ascendingExample 15 Solve the differential equation:Solution:Auxiliary equation is:Page 10

C.F. P.I. P.I Complete solution is:C.F. P.I Example 16 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Page 11

P.IComplete solution is:C.F. P.IExample 17 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Page 12

P.I Complete solution is:C.F. P.ICase IV: WhenUse the rule:Example 18 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Page 13

Complete solution is:C.F. P.IExample 19 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. , Putting, Rationalizing the denominator, Putting P.IComplete solution is:C.F. P.IPage 14

Example 20 Solve the differential equation:Solution:Auxiliary equation is:C.F. P.I. ( P.I.Complete solution is:C.F. P.IPage 15

Example 21 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Imaginary part ofNow P.I. Imaginary part of Page 16

Complete solution is:C.F. P.IExample 22 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. 12 cos2 2 1 sin2 4 14sin2 2 P.I.Complete solution is:C.F. P.ICase V: WhenUse the rule: Page 17

Example 23 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. ,PuttingP.I. Complete solution is:C.F. P.IExample 24 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. Now Page 18

,PuttingAlsoP.I. Complete solution is:C.F. P.ICase VI: Whenis any general function ofmethods I to V aboveResolvenot covered in shortcutinto partial fractions and use the rule:Example 25 Solve the differential equation:Solution: Auxiliary equation is:Page 19

C.F. P.I. , Integrating 2nd term by parts P.I.Complete solution is:C.F. P.IExample 26 Solve the differential equation:Solution: Auxiliary equation is:C.F. P.I. P.I. .①Page 20

Now ② Replacing by – ③Using ②and ③ in ①P.I. P.IComplete solution is:C.F. P.I.Exercise 11ASolve the following differential equations:1.Ans.Page 21

2.Ans.3.Ans.4.Ans.5.Ans.6.Ans.7.8.(1 2tanAns.9.Ans.10., givenwhenAns.11.4 Differential Equations Reducible to Linear Form with ConstantCoefficientsSome special type of homogenous and non homogeneous linear differentialequations with variable coefficients after suitable substitutions can be reducedto linear differential equations with constant coefficients.11.4.1 Cauchy’s Linear Differential EquationThe differential equation of the form:Page 22

is called Cauchy’s linear equation and it can be reduced to linear differentialequations with constant coefficients by following substitutions:, whereSimilarly,and so on.Example 27 Solve the differential equation:, . ①Solution: This is a Cauchy’s linear equation with variable coefficients.Putting,and ① May be rewritten as,Auxiliary equation is:C.F. P.I. , PuttingPuttingPage 23

) ) Complete solution is:C.F. P.IExample 28 Solve the differential equation: . ①Solution: This is a Cauchy’s linear equation with variable coefficients.Putting, ① May be rewritten as,Auxiliary equation is:C.F. P.I. Page 24

PutComplete solution is:C.F. P.I,Example 29 Solve the differential equation:, . ①Solution: This is a Cauchy’s linear equation with variable coefficients.Putting,, ① May be rewritten asPage 25

Auxiliary equation is:C.F. P.I. PuttingComplete solution is:in the 1st termC.F. P.I11.4.2 Legendre’s Linear Differential EquationThe differential equation of the form:is called Legendre’s linear equation and it can be reduced to linear differentialequations with constant coefficients by following substitutions:Page 26

, whereSimilarlyand so on.Example 30 Solve the differential equation: . ①Solution: This is a Legendre’s linear equation with variable coefficients.Putting,Also ① May be rewritten asAuxiliary equation is:C.F. Page 27

P.I. , Puttingcase of failure, alsoin 1st term, it is ain the 2nd term.P.I. Complete solution is:C.F. P.IExample 31 Solve the differential equation: . ①Solution: This is a Legendre’s linear equation with variable coefficients.Putting, ① May be rewritten asAuxiliary equation is:C.F. Page 28

P.I. , Putting, case of failure P.IComplete solution is:C.F. P.I11.5 Method of Variation of Parameters for Finding Particular IntegralMethod of Variation of Parameters enables us to find the solution of 2nd andhigher order differential equations with constant coefficients as well as variablecoefficients.Working ruleConsider a 2nd order linear differential equation: . ①1. Find complimentary function given as: C.F. whereand2. Calculate,are two linearly independent solutions of ①,is called Wronskian of3. Compute,4. Find P.I. 5. Complete solution is given by:andC.F. P.INote: Method is commonly used to solve 2nd order differential but it can beextended to solve differential equations of higher orders.Example 32 Solve the differential equation:Page 29

using method of variation of parameters.Solution:Auxiliary equation is:C.F. andP.I Complete solution is:C.F. P.IExample 33 Solve the differential equation:using method of variation of parameters.Solution: Auxiliary equation is:C.F. andPage 30

P.I Complete solution is:C.F. P.IExample 34 Solve the differential equation:using method of variation of parameters.Solution:Auxiliary equation is:C.F. andP.I Page 31

Complete solution is:C.F. P.IExample 35 Solve the differential equation:using method of variation of parameters.Solution: Auxiliary equation is:C.F. and , PuttingP.IComplete solution is:C.F. P.IPage 32

Example 36 Given thatandsolutions of the differential equation:are two linearly independent,Find the particular integral and general solution using method of variation ofparameters.Solution: Rewriting the equation as:Given thatandC.F. P.IComplete solution is:C.F. P.IExample 37 Solve the differential equation:using method of variation of parameters.Solution: This is a Cauchy’s linear equation with variable coefficients.Putting,Page 33

Given differential equation may be rewritten asAuxiliary equation is:C.F. andP.I Complete solution is:C.F. P.Ior,11.6 Solving Simultaneous Linear Differential EquationsLinear differential equations having two or more dependent variables withsingle independent variable are called simultaneous differential equations andcan be of two types:Page 34

Type 1:,,Consider a system of ordinary differential equations in two dependent variablesand y and an independent variable :,,Given system can be solved as follows:1. Eliminate from the given system of equations resulting a differentialequation exclusively in .2. Solve the differential equation inby usual methods to obtain as afunction of .3. Substitute value ofand its derivatives in one of the simultaneousequations to get an equation in .4. Solve for by usual methods to obtain its value as a function ofExample 38 Solve the system of equations:,Solution: Rewriting given system of differential equations as: ① .②,Multiplying ① by .③Subtracting ② from ③, we get .④which is a linear differential equation inTo solve ④ forwith constant coefficients., Auxiliary equation isC.F. P.I. , Puttingandin 1st and 2nd terms respectively .⑤Page 35

Using ⑤in ① ⑥⑤ and ⑥ give the required solution.Example 39 Solve the system of equations:given that,,Solution: Given system of equations is: ① .②,Multiplying ① by ③Subtracting ② from ③, we get .④which is Cauchy’s linear differential equation inwith variable coefficients.Putting,,④ may be rewritten as ⑤To solve ⑤ for, Auxiliary equation isC.F. Page 36