CHAPTER 1 - FIRST ORDER DIFFERENTIAL EQUATIONS
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTCHAPTER 1FIRST ORDER DIFFERENTIAL EQUATIONSDifferential EquationsDIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0dxd 3x(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0dxd 3x(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0dxd 3x(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0d 3xdx(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0d 3xdx(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTDifferential EquationsIIDefinition: A differential equation is an equation thatcontains a function and one or more of its derivatives. Ifthe function has only one independent variable, then it isan ordinary differential equation. Otherwise, it is apartial differential equation.The following are examples of differential equations: 2u 2u 02 x y 2(b) (x 2 y 2 )dx 2xydy 0d 3xdx(c) x 4xy 0dy 3dy 2 u 2u u2(d) h t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2d xdx(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2d xdx(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2dxd x(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2d xdx(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2d xdx(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTOrder and DegreeIIDefinition: The order of a differential equation is the orderof the highest ordered derivative that appears in the givenequation. The degree of a differential equation is thedegree of the highest ordered derivative treated as avariable.Examples: 2u 2u 0 is of order 2 and degree 1 x 2 y 222(b) (x y )dx 2xydy 0 is of order 1 and degree 1 3 2d xdx(c) x 4xy 0 is of order 3 and degree 2dy 3dy 2 3 u 2u u2(d) h is of order 2 and degree 3 t x 2 y 2(a)DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2x (x 1)dy , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2x (x 1)dy , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2x (x 1)dy , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2xdy (x 1) , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2xdy (x 1) , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of a Differential EquationIIDefinition: A solution of a differential equation is afunction defined explicitly or implicitly by an equation thatsatisfies the given equation. The general solutionrepresents all the possible solutions of the given equation,while a particular solution is any one of the possiblesolutions of a given differential equation.Examples:(a)(b)(c)(d)(e)DIFEQUAdy y , y Cex where C is an arbitrary constantdxdy 3ex , y 3ex C where C is an arbitrary constantdxy (3) 3y 0 2y 0, y e 2xdy (x 1) , (x 1)2 (y 3)2 c 2 , where c is andxy 3arbitrary constant.d 2y k 2 y 0, y sin kt, where k is a constantdt 2DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExistence and Uniqueness TheoremIIThe existence of a particular solution satisfying initialconditions of the form y (x0 ) y0 is guaranteed by thefollowing theorem:Existence and Uniqueness Theorem: Consider a first orderequation of the formdy f (x, y )dxand let T be the rectangular region described byT { (x, y ) R2 x x0 a, y y0 b, a, b are positive constaIf f and fy are continuous in T , then there exists a positivenumber h and a function y y (x) such that(a) y y (x) is a solution of the given equation satisfyingy (x0 ) y0 ; and(b) y y (x) is unique on the interval x x0 h.DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExistence and Uniqueness TheoremIIThe existence of a particular solution satisfying initialconditions of the form y (x0 ) y0 is guaranteed by thefollowing theorem:Existence and Uniqueness Theorem: Consider a first orderequation of the formdy f (x, y )dxand let T be the rectangular region described byT { (x, y ) R2 x x0 a, y y0 b, a, b are positive constaIf f and fy are continuous in T , then there exists a positivenumber h and a function y y (x) such that(a) y y (x) is a solution of the given equation satisfyingy (x0 ) y0 ; and(b) y y (x) is unique on the interval x x0 h.DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExercises(1) Verify that the given function is a solution of the givendifferential equation.(a) y (3) 3y 0 2y 0, y e 2xd 2y k 2 y 0, y sin kt, where k is a constant(b)dt 2(2) Use antiderivatives to obtain a general or a particularsolution to each of the following equations:dy x 3 2xdxdy(b) 4 cos 2xdx(a)DIFEQUAdy 3ex , y 6 x 0dxdy(d) 4y , y 3 whendxx 0(c)DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSeparable Differential EquationIA first order ordinary differential equation can generally beexpressed in the formMdx Ndy 0Iwhere M and N may be functions of two variables x and y .For this reason, we call this the general form of afirst-order ordinary differential equation.If the differential equation can be manipulated algebraicallyto transform it into an equivalent formA(x)dx B(y )dy 0where A(x) is a function of x alone and B(y ) is a functionof y alone, then we say that the variables x and y areseparable.DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSeparable Differential EquationIA first order ordinary differential equation can generally beexpressed in the formMdx Ndy 0Iwhere M and N may be functions of two variables x and y .For this reason, we call this the general form of afirst-order ordinary differential equation.If the differential equation can be manipulated algebraicallyto transform it into an equivalent formA(x)dx B(y )dy 0where A(x) is a function of x alone and B(y ) is a functionof y alone, then we say that the variables x and y areseparable.DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExamplesDIFEQUAIShow that the variables in the differential equation(1 x)y 0 y 2 are separable.IFind the general solution of the equation xyy 0 1 y 2 , ify 3 when x 2.IFind the particular solution of the equation y 0 2xysatisfying y (0) 1.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExamplesDIFEQUAIShow that the variables in the differential equation(1 x)y 0 y 2 are separable.IFind the general solution of the equation xyy 0 1 y 2 , ify 3 when x 2.IFind the particular solution of the equation y 0 2xysatisfying y (0) 1.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExamplesDIFEQUAIShow that the variables in the differential equation(1 x)y 0 y 2 are separable.IFind the general solution of the equation xyy 0 1 y 2 , ify 3 when x 2.IFind the particular solution of the equation y 0 2xysatisfying y (0) 1.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTExercises(1) y 0 y sec x(2) sin x sin y dx cos x cos y dy 0(3) y ln x ln y dx dy 0(4) (xy x) dx (x 2 y 2 x 2 y 2 1) dy(5) dx t(1 t 2 ) sec2 x dt(6) (e2x 4)y 0 y(7) x 2 dx y (x 1) dy 0(8) 2xyy 0 1 y 2 , y 3 when x 2(9) 2y dx 3x dy , y 1 when x 2(10) y 0 x exp(y x 2 ), y 0 when x 0DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTHomogeneous FunctionsIDefinition: Let λ be a parameter. A function f (x, y ) is saidto be homogeneous of degree k , where k is a real number,if f satisfies the conditionf (λx, λy ) λk f (x, y )IDIFEQUAExample: Show that the function f (x, y ) 4x 2 3xy y 2is homogeneous and determine the degree ofhomogeneity.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTHomogeneous FunctionsIDefinition: Let λ be a parameter. A function f (x, y ) is saidto be homogeneous of degree k , where k is a real number,if f satisfies the conditionf (λx, λy ) λk f (x, y )IDIFEQUAExample: Show that the function f (x, y ) 4x 2 3xy y 2is homogeneous and determine the degree ofhomogeneity.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTTwo TheoremsIIDIFEQUATheorem: If M(x, y ) and N(x, y ) are both homogeneousfunctions of the same degree, then the functionM(x, y )is homogeneous of degree zero.g(x, y ) N(x, y )Theorem: If f (x, y ) is homogeneous of degree zero, then fis a function of y /x (or x/y ) alone.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTTwo TheoremsIIDIFEQUATheorem: If M(x, y ) and N(x, y ) are both homogeneousfunctions of the same degree, then the functionM(x, y )is homogeneous of degree zero.g(x, y ) N(x, y )Theorem: If f (x, y ) is homogeneous of degree zero, then fis a function of y /x (or x/y ) alone.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTEquation With Homogeneous CoefficientsDIFEQUAIDefinition: A first order differential equation of the formM dx N dy 0 is said to have homogeneous coefficientsif M and N are homogeneous functions of the samedegree.IExample: Show that the equation3(3x 2 y 2 ) dx 2xy dy 0 has homogeneouscoefficients.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTEquation With Homogeneous CoefficientsDIFEQUAIDefinition: A first order differential equation of the formM dx N dy 0 is said to have homogeneous coefficientsif M and N are homogeneous functions of the samedegree.IExample: Show that the equation3(3x 2 y 2 ) dx 2xy dy 0 has homogeneouscoefficients.DLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of an Equation With HomogeneousCoefficientsGiven: Mdx Ndy 0(a) Transform the equation into the equivalent formg(y /x) dydx 0, where g M/N.(b) Let y vx and use this to convert the second equation intothe form x dv (v g(v )) dx 0 where the variables areseparable.(c) Solve using the method of separation of variables.(d) Use v y /x to obtain a solution in terms of x and y .(e) Remark: You may also use the substitutionx vy , dx v dy y dv .DIFEQUADLSU-Manila
BASIC CONCEPTS SEPARATION OF VARIABLES EQUATIONS WITH HOMOGENEOUS COEFFICIENTS EXACT DIFFERENTSolution of an Equation With HomogeneousCoefficientsGiven: Mdx Ndy 0(a) Transform the equation into the equivalent formg(y /x) dydx 0, where g M/N.(b) Let y vx and use this to convert the second equation intothe form x dv (v g(v )) dx 0 where the variables areseparable.(c) Solve using the method of separation of variables.(d)
I Definition:A differential equation is an equation that contains a function and one or more of its derivatives. If the function has only one independent variable, then it is an ordinary differential equation. Otherwise, it is a partial differential equation. I The following are examples of differential equations: (a) @2u @x2 @2u @y2 0 (b .
Part One: Heir of Ash Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26 Chapter 27 Chapter 28 Chapter 29 Chapter 30 .
TO KILL A MOCKINGBIRD. Contents Dedication Epigraph Part One Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Part Two Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18. Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 Chapter 24 Chapter 25 Chapter 26
DIFFERENTIAL – DIFFERENTIAL OIL DF–3 DF DIFFERENTIAL OIL ON-VEHICLE INSPECTION 1. CHECK DIFFERENTIAL OIL (a) Stop the vehicle on a level surface. (b) Using a 10 mm socket hexagon wrench, remove the rear differential filler plug and gasket. (c) Check that the oil level is between 0 to 5 mm (0 to 0.20 in.) from the bottom lip of the .
DEDICATION PART ONE Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 PART TWO Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 Chapter 21 Chapter 22 Chapter 23 .
DIFFERENTIAL EQUATIONS FIRST ORDER DIFFERENTIAL EQUATIONS 1 DEFINITION A differential equation is an equation involving a differential coefficient i.e. In this syllabus, we will only learn the first order To solve differential equation , we integrate and find the equation y which
Andhra Pradesh State Council of Higher Education w.e.f. 2015-16 (Revised in April, 2016) B.A./B.Sc. FIRST YEAR MATHEMATICS SYLLABUS SEMESTER –I, PAPER - 1 DIFFERENTIAL EQUATIONS 60 Hrs UNIT – I (12 Hours), Differential Equations of first order and first degree : Linear Differential Equations; Differential Equations Reducible to Linear Form; Exact Differential Equations; Integrating Factors .
About the husband’s secret. Dedication Epigraph Pandora Monday Chapter One Chapter Two Chapter Three Chapter Four Chapter Five Tuesday Chapter Six Chapter Seven. Chapter Eight Chapter Nine Chapter Ten Chapter Eleven Chapter Twelve Chapter Thirteen Chapter Fourteen Chapter Fifteen Chapter Sixteen Chapter Seventeen Chapter Eighteen
18.4 35 18.5 35 I Solutions to Applying the Concepts Questions II Answers to End-of-chapter Conceptual Questions Chapter 1 37 Chapter 2 38 Chapter 3 39 Chapter 4 40 Chapter 5 43 Chapter 6 45 Chapter 7 46 Chapter 8 47 Chapter 9 50 Chapter 10 52 Chapter 11 55 Chapter 12 56 Chapter 13 57 Chapter 14 61 Chapter 15 62 Chapter 16 63 Chapter 17 65 .
