Differential Equations - D2cyt36b7wnvt9.cloudfront
DIFFERENTIAL EQUATIONSChapter3799DIFFERENTIAL EQUATIONS He who seeks for methods without having a definite problem in mindseeks for the most part in vain. – D. HILBERT 9.1 IntroductionIn Class XI and in Chapter 5 of the present book, wediscussed how to differentiate a given function f with respectto an independent variable, i.e., how to find f ′(x) for a givenfunction f at each x in its domain of definition. Further, inthe chapter on Integral Calculus, we discussed how to finda function f whose derivative is the function g, which mayalso be formulated as follows:For a given function g, find a function f such thatdy g (x), where y f (x)dx. (1)An equation of the form (1) is known as a differentialequation. A formal definition will be given later.Henri Poincare(1854-1912 )These equations arise in a variety of applications, may it be in Physics, Chemistry,Biology, Anthropology, Geology, Economics etc. Hence, an indepth study of differentialequations has assumed prime importance in all modern scientific investigations.In this chapter, we will study some basic concepts related to differential equation,general and particular solutions of a differential equation, formation of differentialequations, some methods to solve a first order - first degree differential equation andsome applications of differential equations in different areas.9.2 Basic ConceptsWe are already familiar with the equations of the type:x2 – 3x 3 0sin x cos x 0x y 72019-20. (1). (2). (3)
380MATHEMATICSLet us consider the equation:dy y 0. (4)dxWe see that equations (1), (2) and (3) involve independent and/or dependent variable(variables) only but equation (4) involves variables as well as derivative of the dependentvariable y with respect to the independent variable x. Such an equation is called adifferential equation.xIn general, an equation involving derivative (derivatives) of the dependent variablewith respect to independent variable (variables) is called a differential equation.A differential equation involving derivatives of the dependent variable with respectto only one independent variable is called an ordinary differential equation, e.g.,3d 2 y dy . (5)2 2 0 is an ordinary differential equation dx dxOf course, there are differential equations involving derivatives with respect tomore than one independent variables, called partial differential equations but at thisstage we shall confine ourselves to the study of ordinary differential equations only.Now onward, we will use the term ‘differential equation’ for ‘ordinary differentialequation’. Note1. We shall prefer to use the following notations for derivatives:dyd2yd3y y ′ , 2 y ′′, 3 y′′′dxdxdx2. For derivatives of higher order, it will be inconvenient to use so many dashesas supersuffix therefore, we use the notation yn for nth order derivativedny.dx n9.2.1. Order of a differential equationOrder of a differential equation is defined as the order of the highest order derivative ofthe dependent variable with respect to the independent variable involved in the givendifferential equation.Consider the following differential equations:dy exdx2019-20. (6)
DIFFERENTIAL EQUATIONSd2y y 0dx 2381. (7)3 d3y 2 d 2y 3 x 2 0 dx dx . (8)The equations (6), (7) and (8) involve the highest derivative of first, second andthird order respectively. Therefore, the order of these equations are 1, 2 and 3 respectively.9.2.2 Degree of a differential equationTo study the degree of a differential equation, the key point is that the differentialequation must be a polynomial equation in derivatives, i.e., y′, y″, y″′ etc. Consider thefollowing differential equations:2d 3 y d 2 y dy 2 2 y 0dx3 dx dx. (9)2 dy dy 2 sin y 0dxdx . (10)dy dy sin 0dx dx . (11)We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)is a polynomial equation in y′ (not a polynomial in y though). Degree of such differentialequations can be defined. But equation (11) is not a polynomial equation in y′ anddegree of such a differential equation can not be defined.By the degree of a differential equation, when it is a polynomial equation inderivatives, we mean the highest power (positive integral index) of the highest orderderivative involved in the given differential equation.In view of the above definition, one may observe that differential equations (6), (7),(8) and (9) each are of degree one, equation (10) is of degree two while the degree ofdifferential equation (11) is not defined. NoteOrder and degree (if defined) of a differential equation are alwayspositive integers.2019-20
382MATHEMATICSExample 1 Find the order and degree, if defined, of each of the following differentialequations:(i)(iii)2d2ydy dy 0(ii) xy 2 x ydx dx dxdy cos x 0dxy ′′′ y 2 e y′ 0Solution(i) The highest order derivative present in the differential equation isdy, so itsdxorder is one. It is a polynomial equation in y′ and the highest power raised todydxis one, so its degree is one.d2y(ii) The highest order derivative present in the given differential equation is 2 , sodxits order is two. It is a polynomial equation inpower raised todyd2yand the highest2 anddxdxd2yis one, so its degree is one.dx 2(iii) The highest order derivative present in the differential equation is y′′′ , so itsorder is three. The given differential equation is not a polynomial equation in itsderivatives and so its degree is not defined.EXERCISE 9.1Determine order and degree (if defined) of differential equations given in Exercises1 to 10.1.d4y sin( y′′′) 0dx442. y′ 5y 0d 2s ds 3. 3s 2 0 dt dt2 d2y dy 4. 2 cos 0 dx dx 5.6. ( y′′′) 2 (y″)3 (y′)4 y5 07. y ′′′ 2y″ y′ 02019-20d2y cos3x sin 3xdx 2
DIFFERENTIAL EQUATIONS3838. y′ y ex9. y″ (y′)2 2y 0 10. y″ 2y′ sin y 011. The degree of the differential equation3 d 2 y dy 2 dy 2 sin 1 0 is dx dx dx (A) 3(B) 2(C) 112. The order of the differential equationd2ydy 3 y 0 is2dxdx(A) 2(B) 1(D) not defined2x 2(C) 0(D) not defined9.3. General and Particular Solutions of a Differential EquationIn earlier Classes, we have solved the equations of the type:. (1)x2 1 02. (2)sin x – cos x 0Solution of equations (1) and (2) are numbers, real or complex, that will satisfy thegiven equation i.e., when that number is substituted for the unknown x in the givenequation, L.H.S. becomes equal to the R.H.S.d2y y 0. (3)dx 2In contrast to the first two equations, the solution of this differential equation is afunction φ that will satisfy it i.e., when the function φ is substituted for the unknown y(dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S.The curve y φ (x) is called the solution curve (integral curve) of the givendifferential equation. Consider the function given byy φ (x) a sin (x b),. (4)where a, b R. When this function and its derivative are substituted in equation (3),L.H.S. R.H.S. So it is a solution of the differential equation (3).Now consider the differential equationLet a and b be given some particular values say a 2 and b π, then we get a4π y φ1(x) 2sin x . (5) 4 When this function and its derivative are substituted in equation (3) againL.H.S. R.H.S. Therefore φ1 is also a solution of equation (3).function2019-20
384MATHEMATICSFunction φ consists of two arbitrary constants (parameters) a, b and it is calledgeneral solution of the given differential equation. Whereas function φ1 contains noarbitrary constants but only the particular values of the parameters a and b and henceis called a particular solution of the given differential equation.The solution which contains arbitrary constants is called the general solution(primitive) of the differential equation.The solution free from arbitrary constants i.e., the solution obtained from the generalsolution by giving particular values to the arbitrary constants is called a particularsolution of the differential equation.Example 2 Verify that the function y e– 3x is a solution of the differential equationd 2 y dy 6y 0dx 2 dxSolution Given function is y e– 3x. Differentiating both sides of equation with respectto x , we getdy 3 e 3 xdxNow, differentiating (1) with respect to x, we have. (1)d2y 9 e – 3xdx 2d 2 y dy, and y in the given differential equation, we getdx 2 dxL.H.S. 9 e– 3x (–3e– 3x) – 6.e– 3x 9 e– 3x – 9 e– 3x 0 R.H.S.Therefore, the given function is a solution of the given differential equation.Substituting the values ofExample 3 Verify that the function y a cos x b sin x, where, a, b R is a solution2of the differential equation d y y 0dx 2Solution The given function isy a cos x b sin x. (1)Differentiating both sides of equation (1) with respect to x, successively, we getdy – a sin x b cos xdxd2y – a cos x – b sin xdx 22019-20
DIFFERENTIAL EQUATIONS385d2yand y in the given differential equation, we getdx 2L.H.S. (– a cos x – b sin x) (a cos x b sin x) 0 R.H.S.Therefore, the given function is a solution of the given differential equation.Substituting the values ofEXERCISE 9.2In each of the Exercises 1 to 10 verify that the given functions (explicit or implicit) is asolution of the corresponding differential equation:: y″ – y′ 01. y ex 12: y′ – 2x – 2 02. y x 2x C3. y cos x C: y′ sin x 05. y Ax:xy1 x2xy′ y (x 0)6. y x sin x:xy′ y x4. y 1 x2:7. xy log y C:8. y – cos y x9. x y tan–1y::y′ x 2 y 2 (x 0 and x y or x – y)y2y′ (xy 1)1 xy(y sin y cos y x) y′ yy2 y′ y2 1 0dy 0 (y 0)dx11. The number of arbitrary constants in the general solution of a differential equationof fourth order are:(A) 0(B) 2(C) 3(D) 412. The number of arbitrary constants in the particular solution of a differential equationof third order are:(A) 3(B) 2(C) 1(D) 010. y a 2 x 2 x (–a, a) :x y9.4 Formation of a Differential Equation whose General Solution is givenWe know that the equationx2 y2 2x – 4y 4 0represents a circle having centre at (– 1, 2) and radius 1 unit.2019-20. (1)
386MATHEMATICSDifferentiating equation (1) with respect to x, we getx 1dy (y 2)2 ydx. (2)which is a differential equation. You will find later on [See (example 9 section 9.5.1.)]that this equation represents the family of circles and one member of the family is thecircle given in equation (1).Let us consider the equation. (3)x2 y2 r 2By giving different values to r, we get different members of the family e.g.x2 y2 1, x2 y2 4, x2 y2 9 etc. (see Fig 9.1).Thus, equation (3) represents a family of concentriccircles centered at the origin and having different radii.We are interested in finding a differential equationthat is satisfied by each member of the family. Thedifferential equation must be free from r because r isdifferent for different members of the family. Thisequation is obtained by differentiating equation (3) withrespect to x, i.e.,2x 2ydy 0dxorx ydy 0dx. (4)Fig 9.1which represents the family of concentric circles given by equation (3).Again, let us consider the equationy mx c. (5)By giving different values to the parameters m and c, we get different members ofthe family, e.g.,y x(m 1, c 0)y 3x(m 3 , c 0)y x 1(m 1, c 1)y –x(m – 1, c 0)y –x–1(m – 1, c – 1) etc.( see Fig 9.2).Thus, equation (5) represents the family of straight lines, where m, c are parameters.We are now interested in finding a differential equation that is satisfied by eachmember of the family. Further, the equation must be free from m and c because m and2019-20
DIFFERENTIAL EQUATIONSc are different for different members of the family.This is obtained by differentiating equation (5) withrespect to x, successively we gety y 3xYy –x3871x y xy –x–1dyd2y m , and 0dxdx 2. (6) X’XOThe equation (6) represents the family of straightlines given by equation (5).Note that equations (3) and (5) are the generalsolutions of equations (4) and (6) respectively.Y’Fig 9.29.4.1 Procedure to form a differential equation that will represent a givenfamily of curves(a) If the given family F 1 of curves depends on only one parameter then it isrepresented by an equation of the formF1 (x, y, a) 0. (1)For example, the family of parabolas y2 ax can be represented by an equationof the form f (x, y, a) : y2 ax.Differentiating equation (1) with respect to x, we get an equation involvingy′, y, x, and a, i.e.,g (x, y, y′, a) 0. (2)The required differential equation is then obtained by eliminating a from equations(1) and (2) asF (x, y, y′) 0. (3)(b) If the given family F2 of curves depends on the parameters a, b (say) then it isrepresented by an equation of the fromF2 (x, y, a, b) 0. (4)Differentiating equation (4) with respect to x, we get an equation involvingy′, x, y, a, b, i.e.,g (x, y, y′, a, b) 0. (5)But it is not possible to eliminate two parameters a and b from the two equationsand so, we need a third equation. This equation is obtained by differentiatingequation (5), with respect to x, to obtain a relation of the formh (x, y, y′, y″, a, b) 02019-20. (6)
388MATHEMATICSThe required differential equation is then obtained by eliminating a and b fromequations (4), (5) and (6) asF (x, y, y′, y″) 0. (7) NoteThe order of a differential equation representing a family of curves issame as the number of arbitrary constants present in the equation corresponding tothe family of curves.Example 4 Form the differential equation representing the family of curves y mx,where, m is arbitrary constant.Solution We havey mx. (1)Differentiating both sides of equation (1) with respect to x, we getdy mdxdy xSubstituting the value of m in equation (1) we get y dxdyxor–y 0dxwhich is free from the parameter m and hence this is the required differential equation.Example 5 Form the differential equation representing the family of curvesy a sin (x b), where a, b are arbitrary constants.Solution We havey a sin (x b). (1)Differentiating both sides of equation (1) with respect to x, successively we getdy a cos (x b)dxd2y – a sin (x b)dx 2Eliminating a and b from equations (1), (2) and (3), we get. (2). (3)d2y y 0. (4)dx 2which is free from the arbitrary constants a and b and hence this the required differentialequation.2019-20
DIFFERENTIAL EQUATIONS389Example 6 Form the differential equationrepresenting the family of ellipses having foci onx-axis and centre at the origin.Solution We know that the equation of said familyof ellipses (see Fig 9.3) isx2 y 2 1a 2 b2. (1)Fig 9.3Differentiating equation (1) with respect to x, we get2 x 2 y dy 0a 2 b 2 dx2y dy b 2x dx aor. (2)Differentiating both sides of equation (2) with respect to x, we get y x dy d 2 y x dx y dy 0 2 dxx 2 dx 0or. (3)which is the required differential equation.Example 7 Form the differential equation of the familyof circles touching the x-axis at origin.YSolution Let C denote the family of circles touchingx-axis at origin. Let (0, a) be the coordinates of thecentre of any member of the family (see Fig 9.4).Therefore, equation of family C isx2 (y – a)2 a2 or x2 y2 2ay. (1)where, a is an arbitrary constant. Differentiating bothsides of equation (1) with respect to x,we get2x 2 ydydy 2adxdx2019-20X’OY’Fig 9.4X
390orMATHEMATICSdydyx y aor a dxdxx ydydx. (2)dydxSubstituting the value of a from equation (2) in equation (1), we getdy x y dx x2 y2 2ydydxdy 2dy( x y 2 ) 2 xy 2 y 2dxdxordy2 xy 2 2dxx –yorThis is the required differential equation of the given family of circles.Example 8 Form the differential equation representing the family of parabolas havingvertex at origin and axis along positive direction of x-axis.Solution Let P denote the family of above said parabolas (see Fig 9.5) and let (a, 0) be thefocus of a member of the given family, where a is an arbitrary constant. Therefore, equationof family P isy2 4ax. (1)Differentiating both sides of equation (1) with respect to x, we getdy 4adxSubstituting the value of 4a from equation (2)in equation (1), we get2y. (2)dy y2 2 y ( x)dx ory 2 2 xydy 0dxwhich is the differential equation of the given familyof parabolas.2019-20Fig 9.5
DIFFERENTIAL EQUATIONS391EXERCISE 9.3In each of the Exercises 1 to 5, form a differential equation representing the givenfamily of curves by eliminating arbitrary constants a and b.1.4.6.7.8.9.10.11.x y 12. y2 a (b2 – x2)3. y a e3x b e– 2xa by e2x (a bx)5. y ex (a cos x b sin x)Form the differential equation of the family of circles touching the y-axis atorigin.Form the differential equation of the family of parabolas having vertex at originand axis along positive y-axis.Form the differential equation of the family of ellipses having foci on y-axis andcentre at origin.Form the differential equation of the family of hyperbolas having foci on x-axisand centre at origin.Form the differential equation of the family of circles having centre on y-axisand radius 3 units.Which of the following differential equations has y c1 ex c2 e–x as the generalsolution?d2yd2yd2yd2y 0 0 1 0 1 0yy(B)(C)(D)dx 2dx 2dx 2dx 212. Which of the following differential equations has y x as one of its particularsolution?(A)(A)d2ydy x2 xy x2dxdx(B)d2ydy x xy x2dxdx(C)d 2 y 2 dy x xy 0dxdx2(D)d2ydy x xy 02dxdx9.5. Methods of Solving First Order, First Degree Differential EquationsIn this section we shall discuss three methods of solving first order first degree differentialequations.9.5.1 Differential equations with variables separableA first order-first degree differential equation is of the formdy F (x, y)dx2019-20. (1)
392MATHEMATICSIf F (x, y) can be expressed as a product g (x) h(y), where, g(x) is a function of xand h(y) is a function of y, then the differential equation (1) is said to be of variableseparable type. The differential equation (1) then has the formdy h (y) . g (x). (2)dxIf h (y) 0, separating the variables, (2) can be rewritten as1dy g (x) dxh( y ). (3)Integrating both sides of (3), we get1 h( y) dy g ( x) dx. (4)Thus, (4) provides the solutions of given differential equation in the formH (y) G (x) C1Here, H (y) and G (x) are the anti derivatives of h ( y ) and g (x) respectively andC is the arbitrary constant.Example 9 Find the general solution of the differential equationdy x 1 , (y 2)dx 2 ySolution We havex 1dy 2 ydx. (1)Separating the variables in equation (1), we get(2 – y) dy (x 1) dxIntegrating both sides of equation (2), we get (2 y) dy ( x 1) dxororor2y y2x2 x C1 22x2 y2 2x – 4y 2 C1 0x2 y2 2x – 4y C 0, where C 2C1which is the general solution of equation (1).2019-20. (2)
DIFFERENTIAL EQUATIONSExample 10 Find the general solution of the differential equation393dy 1 y 2. dx 1 x 2Solution Since 1 y2 0, therefore separating the variables, the given differentialequation can be written asdydx2 1 y1 x2Integrating both sides of equation (1), we getdy 1 y2 . (1)dx 1 x2ortan–1 y tan–1x Cwhich is the general solution of equation (1).Example 11 Find the particular solution of the differential equationdy 4 xy 2 givendxthat y 1, when x 0.Solution If y 0, the given differential equation can be written asdy – 4x dxy2Integrating both sides of equation (1), we getdy y2or 4 x dx1 – 2x2 Cy12x CSubstituting y 1 and x 0 in equation (2), we get, C – 1.or. (1)y 2. (2)Now substituting the value of C in equation (2), we get the particular solution of thegiven differential equation as y 12x 12.Example 12 Find the equation of the curve passing through the point (1, 1) whosedifferential equa
In contrast to the first two equations, the solution of this differential equation is a function φ that will satisfy it i.e., when the function φ is substituted for the unknown y (dependent variable) in the given differential equation, L.H.S. becomes equal to R.H.S.
(iii) introductory differential equations. Familiarity with the following topics is especially desirable: From basic differential equations: separable differential equations and separa-tion of variables; and solving linear, constant-coefficient differential equations using characteristic equations.
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 .
3 Ordinary Differential Equations K. Webb MAE 4020/5020 Differential equations can be categorized as either ordinary or partialdifferential equations Ordinarydifferential equations (ODE's) - functions of a single independent variable Partial differential equations (PDE's) - functions of two or more independent variables
Chapter 1 Introduction 1 1.1 ApplicationsLeading to Differential Equations 1.2 First Order Equations 5 1.3 Direction Fields for First Order Equations 16 Chapter 2 First Order Equations 30 2.1 Linear First Order Equations 30 2.2 Separable Equations 45 2.3 Existence and Uniqueness of Solutionsof Nonlinear Equations 55
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
Introduction to Advanced Numerical Differential Equation Solving in Mathematica Overview The Mathematica function NDSolve is a general numerical differential equation solver. It can handle a wide range of ordinary differential equations (ODEs) as well as some partial differential equations (PDEs). In a system of ordinary differential equations there can be any number of
13.1 Differential Equations and Laplace Transforms 189 13.2 Discontinuous Functions 192 13.3 Differential Equations with Discontinuous Forcing 194 Problem Set E: Series Solutions and Laplace Transforms 197 14 Higher Order Equations and Systems of First Order Equations 211 14.1 Higher Order Linear Equations 212
Wei-Chau Xie is a Professor in the Department of Civil and Environmental Engineering and the Department of Applied Mathematics at the University of Waterloo. He is the author of Dynamic Stability of Structures and has published numerous journal articles on dynamic stability, structural dynamics and random vibration, nonlinear dynamics and stochastic mechanics, reliability and safety analysis .
