DIFFERENTIAL EQUATIONS203 - PDF document

DIFFERENTIAL EQUATIONS203 (iv)Correct substitution for the solution of the differential equation of thetype (,)dxgxydy where () is a homogeneous function of thedegree zero is =vy.(v)Number of arbitrary constants in the particular solution of a differentialequation of order two is two.(vi)The differential equation representing the family of circles+ ( = will be of order two.(vii)The solution of dyydxxæöç÷èø is 23y ‚ 23x (viii)Differential equation representing the family of curves (Acos + Bsin) is ‚220dydydxdx+= (ix)The solution of the differential equation dyxydxx is + =kx(x)Solution of tanxdyyyxdxx=+ is sincxæöç÷èø (xi)The differential equation of all non horizontal lines in a plane is dxdy . 202 MATHEMATICS (iii)The number of arbitrary constants in the general solution of a differentialequation of order three is _________.(iv) 1 logdyy dxxxx += is an equation

of the type _________.(v)General solution of the differential equation of the type 11 PQ dxdy += is given by _________.(vi)The solution of the differential equation 2 xdy yx dx += is _________.(vii)The solution of (1 + x2) dy dx +2xy – 4x2 = 0 is _________.(viii)The solution of the differential equation ydx + (x + xy)dy = 0 is ______.(ix)General solution of dy y dx + = sinx is _________.(x)The solution of differential equation coty dx = xdy is _________.(xi)The integrating factor of 1 dyy y dxx + += is _________.77.State True or False for the following:(i)Integrating factor of the differential of the form 11 Q dxpxdy += is givenby 1 pdy e
.(ii)Solution of the differential equation of the type 11 Q dxpxdy += is givenby x.I.F. = (I.F)Q dy ´ .(iii)Correct substitution for the solution of the differential equation of thetype (,) dy fxy dx , where f (x, ) is a hom

ogeneous function of zerodegree is y = vx. DIFFERENTIAL EQUATIONS201 71.General solution of tansecdyyxxdx+= is :(A)sec = tan +(B)tan = sec +(C) tan =tan +(D)sec = tan +Solution of the differential equation sindyydxx+= is :(A)+ cos) = sin +(B)‚ cos) = sin +(C)xycos = sin +(D)+ cos) = cos +The general solution of the differential equation ( + 1)ydy = ( + 1) is:(A) (+ 1)= + 1)(B)+ 1= + 1 +(C)= log { + 1) ( + 1)}(D) logìü=+íýîþ The solution of the differential equation dydx + x e€yis :(A) +(B) 33x (C) 33x (D) 33x The solution of the differential equation 222211(1)dyxydxxx+=++ is :(A) (1 +) = + tan‚1(B) 21yx+ = c + tan‚1(C) log (1 +) = + tan‚1(D) (1 +) = + sin‚1Fill in the blanks of the following (i to xi)(i)The degree of the differential equation dydxdydx+= is _________.(ii)The degree of the differential equation dydxæö+=ç÷èø is _________. 200

MATHEMATICS The differential equation for which =cos +sin is a solution, is :(A) dydx = 0(B) dydx = 0(C) dydx + ( b = 0(D) dydx + ( b = 0The solution of dydx = (0) = 0 is :(A) = ‚ 1)(B) xe(C)xe + 1(D) xe€xThe order and degree of the differential equation 32dydydydxdxdxæöæö-+=ç÷ç÷èøèø are :(A) 1, 4(B) 3, 4(C) 2, 4(D) 3, 2The order and degree of the differential equation dydydxdxéùæö+=êúç÷èøêúëû are :(A) 2, 32 (B) 2, 3(C) 2, 1(D) 3, 4The differential equation of the family of curves = 4 +) is :(A) dydyyxdxdxæö=+ç÷èø (B) 24ya (C) dydydxdxæö+=ç÷èø (D) dydyxyydxdxæöç÷èø Which of the following is the general solution of 20dydydxdx-+= (A) = (A + B)e(B) = (A + B)(C) = A + B(D) = Acos + Bsin DIFFERENTIAL EQUATIONS199 59.The differential equation of the family of curves + ‚ 2= 0, where isarbitrary constant, is:(A) ( ‚ dydx = 2xy(B

) 2 ( + dydx xy(C) 2 ( ‚ dydx xy(D) ( + dydx =xyFamily = A + A of curves will correspond to a differential equation of order(A) 3(B) 2(C) 1(D) not definedThe general solution of dydx = 2 xy is :(A) xy (B) 2xe (C) 2xe (D) 2xe The curve for which the slope of the tangent at any point is equal to the ratio ofthe abcissa to the ordinate of the point is :(A) an ellipse(B) parabola(C) circle(D) rectangular hyperbolaThe general solution of the differential equation 22xdyedx= xyis :(A) yce (B) yce (C) ()yxce=+ (D) ()ycxe=- The solution of the equation (2 ‚ 1) dx ‚ (2 + 3) = 0 is :(A) 2123 (B) 2123 (C) 2321 (D) 2121 198 MATHEMATICS 53.Integrating factor of the differential equation tan‚sec0dyyxxdx is:(A) cos(B) sec(C)cos(D)sec54.The solution of the differential equation dyydx is:(A)= tan‚1(B)y € x k (1 +xy(C)= tan‚1(D) tan(xy)=55.The integrating factor of th

e differential equation dyydxx+= is:(A) xxe (B) xex (C)xe(D)56. =mxmx satisfies which of the following differential equation?(A) dymydx+= (B) dymydx-= (C) dymydx-= (D) dymydx+= 57.The solution of the differential equation cos sinydx + sin cosydy = 0 is :(A) sinsin (B) sinsin(C) sin+ sin(D) coscos c58.The solution of dydx = is:(A)y = ekxx (B)xe +cx(C)xe +(D)x = ekyy DIFFERENTIAL EQUATIONS197 45.The number of solutions of dyydxx when (1) = 2 is :(A) none(B) one(C) two(D) infiniteWhich of the following is a second order differential equation?(A) ( =(B)¢¢ + = sin(C)¢¢¢ + (¢¢2 = 0(D)Integrating factor of the differential equation (1 ‚ dyxydx-= is(A) ‚(B) 21xx+ (C) 21x- (D) 12 log (1 ‚ xtan‚1 + tan‚1 = is the general solution of the differential equation:(A) dyydx (B) dyxdx (C) (1 + + (1 + = 0(D) (1 + + (1 + = 0The differential equation dydx = represents

:(A) Family of hyperbolas(B) Family of parabolas(C) Family of ellipses(D) Family of circlesThe general solution ofcosy dxsiny dy = 0 is :(A)cos k(B)sin k(C) = kcos(D) = ksinThe degree of the differential equation 60dydydxdxæö++=ç÷èø is :(A) 1(B) 2(C) 3(D) 5The solution of ,(0)0dyyeydx+== is :(A) e ‚ 1)(B) xe€x(C) xe€x+ 1(D)=( + 1)€x 196 MATHEMATICS 38.The differential equation for = Acos + Bsin, where A and B are arbitraryconstants is(A) dydx-a= (B) dydx+a= (C) dydx+a= (D) dydx-a= Solution of differential equationxdy ‚ydx = 0 represents :(A) a rectangular hyperbola(B) parabola whose vertex is at origin(C) straight line passing through origin(D) a circle whose centre is at originIntegrating factor of the differential equation cos dydx sin = 1 is :(A) cos(B) tan(C) sec(D) sinSolution of the differential equation tany secxdx + tan secydy = 0 is :(A) tan+ tan

=(B) tan‚ tan =(C) tantan (D) tan. tan =Family = A + A of curves is represented by the differential equation of degree(A) 1(B) 2(C) 3 (D) 4Integrating factor of xdydx ‚ 3 is :(A)(B) log(C) 1x (D)€ xSolution of dydx-= , y(0) = 1is given by(A)xy = ‚(B)xy = ‚(C)xy = ‚ 1(D) = 2 ‚ 1 DIFFERENTIAL EQUATIONS195 Find the equation of the curve through the point (1, 0) if the slope of the tangentto the curve at any point () is xx Find the equation of a curve passing through origin if the slope of the tangent tothe curve at any point () is equal to the square of the difference of the abcissaand ordinate of the point.Find the equation of a curve passing through the point (1, 1). If the tangentdrawn at any point P () on the curve meets the co-ordinate axes at A and Bsuch that P is the mid-point of AB.Solve : dyxydx (log ‚ log + 1)Objective TypeChoose the cor

rect answer from the given four options in each of the Exercises from34 to 75 (M.C.Q)The degree of the differential equation sinis:dydydydxdxdxæöæöæö+=ç÷ç÷ç÷èøèøèø (A) 1(B) 2(C) 3(D) not definedThe degree of the differential equation isdydyéùæö+=êúç÷èøêúëû (A) 4(B) 32 (C) not defined(D) 2The order and degree of the differential equation +0dydydxdxæö+=ç÷èø respectively, are(A) 2 and not defined(B) 2 and 2(C) 2 and 3 (D) 3 and 3If = (Acos + Bsin), then is a solution of(A) 20dydydxdx+= (B) 220dydydxdx-+= (C) 220dydydxdx++= (D) 20dydx+= 194 MATHEMATICS 13.Form the differential equation having = (sin‚1 + Acos‚1 + B, where A and Bare arbitrary constants, as its general solution.Form the differential equation of all circles which pass through origin and whosecentres lie on-axis.Find the equation of a curve passing through origin and satisfying th

e differentialequation (1)24dyxxyxdx++= Solve : dydx =xy +Find the general solution of the differential equation (1 +) + ( ‚tan‚1 dydx = 0.Find the general solution of + ( ‚xy + = 0.Solve : ( +) ( ‚) = +dy..Hint: Substitute + = after seperatingand dySolve : 2 ( + 3) ‚xy dydx = 0, given that (1) = ‚ 2.Solve the differential equation = cos (2 ‚ ycosec given that = 2 when 2xp= Form the differential equation by eliminating A and B in A + B = 1.Solve the differential equation (1 +) tan‚1xdx + 2 (1 + = 0.Find the differential equation of system of concentric circles with centre (1, 2).Long Answer (L.A.)Solve : ()yxydx = (sin + logFind the general solution of (1 + tan) ( ‚) + 2xdy = 0.Solve : dydx = cos( +) + sin ( +).[Hint: Substitute + =Find the general solution of 3sin2dyyxdx-= Find the equation of a curve passing through (2, 1) if the slope of

the tangent tothe curve at any point () is 22xyxy . DIFFERENTIAL EQUATIONS193 (x)False, because = does not satisfy the given differential equation.9.3EXERCISEShort Answer (S.A.)1.Find the solution of yxdydx Find the differential equation of all non vertical lines in a plane.Given that dydx and = 0 when = 5.Find the value of when = 3.Solve the differential equation ( ‚ 1) dydx + 2xy = 211x- Solve the differential equation dyxyydx+= Find the general solution of mxdyayedx+= Solve the differential equation xydydx+= Solve:ydx ‚xdy =ydxSolve the differential equation dydx = 1 + + +xy, when = 0, = 0.Find the general solution of ( + 2 dydx 11.If) is a solution of 2sinxdyydxæöç÷èø = ‚ cos and y (0) = 1, then find the value æöç÷èø If) is a solution of (1 + dydt ty = 1 and (0) = ‚ 1, then show that (1) = ‚ 12 . 192 MATHEMATICS (x) =is a particular solution o

f the differential equation dydyxxyxdxdx-+= Solution(i)True, since the equation representing the given family is 2222xyab+= , whichhas two arbitrary constants.(ii)True, because it is not a polynomial equation in its derivatives.(iii)True(iv)True, because () =).(v)True, because () =).(vi)False, because I.F = dxee (vii)True, because given equation can be written as 2211xydxdyxy++ log (1 +) = ‚ log (1 +) + log(1 +) (1 +) =(viii)False, since I.F. = seclog(sectan)xdxxxee = sec + tan,the solution is, (sec + tan) = (sectan)tanxxxdx = sec tan + sec1xxxdx =sec + tanx+k(ix)True,y=tan‚1 dydydxdx+= Þ ‚11dydxæöç÷èø , i.e., (1)dyydx-+ which satisfies the given equation. DIFFERENTIAL EQUATIONS191 dydx = ‚2eyxx i.e. dydx + yx = ‚2xex This is a differential equation of the type dydx + P = QExample 23State whether the following statements areTrue orFalse(i)Order of t

he differential equation representing the family of ellipses havingcentre at origin and foci on-axis is two.(ii)Degree of the differential equation dy x+ dydx is not defined.(iii) dydx+= is a differential equation of the type PQdydx+= but it can be solvedusing variable separable method also.(iv)F() = coscosyxæöç÷èøæöç÷èø is not a homogeneous function.(v)F() = 22xyxy is a homogeneous function of degree 1.(vi)Integrating factor of the differential equation cosdyyxdx-= is(vii)The general solution of the differential equation(1 + + (1 + = 0is (1 +) (1 +) =(viii)The general solution of the differential equation secyx = tan is (sec ‚ tan) = sec ‚ tan + +(ix) + = tan‚1 is a solution of the differential equation 10dydx++= 190 MATHEMATICS (viii)The general solution of the differential equation dyydxx+= is __________ .(ix)The differential equation representing th

e family of curves = A sin + Bcos is __________ .(x) 1(0)eydxdyxxæö-=¹ç÷èø when written in the form PQdydx+= , thenP = __________ .Solution(i)One; is the only arbitrary constant.(ii)Two; since the degree of the highest order derivative is two.(iii)Zero; any particular solution of a differential equation has no arbitrary constant.(iv)Zero.(v) =vy(vi) 1x ; given differential equation can be written as sindyyxdxxx-= and thereforeI.F. = 1dxxe-ò ‚log 1x (vii) +from given equation, we have =dx.(viii)xy = 22xc+ ; I.F. = dx =log =and the solution is . = .1xdx = 2C2x+ (ix) 0;dydx+= Differentiating the given function w.r.t. successively, we get dydx = Acos ‚ Bsinand dydx = ‚Asin ‚ Bcos dydx + = 0 is the differential equation.(x) 1x ; the given equation can be written as DIFFERENTIAL EQUATIONS189 Example 21The solution of the differential equation dyxyxdx+= is(A)

xc (B) yc=+ (C) xc (D) xc SolutionCorrect answer is (D). I.F. = 2loglog2dxxxeeex=== . Therefore, the solutionis = xxdxk=+ , i.e., = xc Example 22Fill in the blanks of the following:(i)Order of the differential equation representing the family of parabolas = 4 is __________ .(ii)The degree of the differential equation dydydxdxæöæö+=ç÷ç÷èøèø is ________ .(iii)The number of arbitrary constants in a particular solution of the differentialequation tanx dx+ tany dy= 0 is __________ .(iv)F () = 22xyy++ is a homogeneous function of degree__________ .(v) An appropriate substitution to solve the differential equation dxdy = loglogxxxyæöç÷èøæöç÷èø is__________ .(vi)Integrating factor of the differential equation dyxydx = sin is __________ .(vii)The general solution of the differential equation xydydx is __________ . 188 MATHEMATICS SolutionCorrect answer is (C). Give

n equation can be written as dydxyx Þ 2log ( + 3) = log + log + 3)cxwhich represents the family of parabolasExample 17The integrating factor of the differential equation dydx log) + = 2log is(A)(B) log(C) log (log)(D)SolutionCorrect answer is (B). Given equation can be written as logdyydxxxx+= Therefore,I.F. = logdxxx =log(log = logExample 18A solution of the differential equation dydyxydxdxæö-+=ç÷èø is(A)= 2(B)= 2(C)= 2 ‚ 4(D)= 2 ‚ 4Solution Correct answer is (C).Example 19Which of the following is not a homogeneous function of and(A) + 2xy(B) 2 ‚(C) cosyyxxæöç÷èø (D) sin‚ cosSolution Correct answer is (D).Example 20Solution of the differential equation dxdyxy+= is(A) 11xy+= (B) logx .log c(C) xy(D)Solution Correct answer is (C). From the given equation, we get log + log = loggivingxy = DIFFERENTIAL EQUATIONS187 Objective Type QuestionsChoose th

e correct answer from the given four options in each of theExamples 12 to 21.Example 12The degree of the differential equation dydydxdxæöæö+=ç÷ç÷èøèø is(A) 1(B) 2(C) 3(D) 4SolutionThe correct answer is (B).Example 13The degree of the differential equation 3logdydydyæöæö+=ç÷ç÷èøèø is(A) 1(B) 2(C) 3(D) not definedSolutionCorrect answer is (D). The given differential equation is not a polynomialequation in terms of its derivatives, so its degree is not defined.Example 14The order and degree of the differential equation dydyéùæö+=êúç÷èøêúëû respectively, are(A) 1, 2(B) 2, 2(C) 2, 1 (D) 4, 2SolutionCorrect answer is (C).Example 15The order of the differential equation of all circles of given radiusis:(A) 1(B) 2(C) 3(D) 4SolutionCorrect answer is (B). Let the equation of given family be + ( =. It has two orbitrary constants and. Threrefore, the order ofthe

given differential equation will be 2.Example 16The solution of the differential equation 2.‚xy = 3 represents a family of(A) straight lines (B) circles(C) parabolas(D) ellipses 186 MATHEMATICS Substituting = 1, = 2p , we get = 32 , therefore, 13tanæö=-+ç÷èø is the required solution.Example 11State the type of the differential equation for the equation.xdy ‚ydx = 22xy and solve it.SolutionGiven equation can be written asxdy 22xyydx++ ,i.e., 22xyydxx++ ... (1)Clearly RHS of (1) is a homogeneous function of degree zero. Therefore, the givenequation is a homogeneous differential equation. Substituting =vx, we get from (1) 222xvxvxdvvxdxx+++= i.e. dvvxvvdx+=++ dvxvdx=+ Þ dvdx ... (2)Integrating both sides of (2), we getlog ( 21v+ ) = log + logÞ + 21v+ =cxÞ yx + 221yx+ =cxÞ y + 22xy =cx DIFFERENTIAL EQUATIONS185 secvdv = dx tan= € logx ctan logxc+= ..

.(ii) Substituting = 1, = 4p , we get. = 1. Thus, we gettan æöç÷èø + log = 1, which is the required equation.Example 10Solve dyxxydx = 1 + cos æöç÷èø 0 and = 1, = 2p SolutionGiven equation can be written as dyxxydx = 2cos æöç÷èø 0. 2cosdyxxydxæöç÷èø Þ secdyxxydxæöç÷èøéù-=êúëû Dividing both sides by , we get 23secdyxydxxxæöéùç÷êúèøêúêúëû Þ tandydxxéùæöç÷êúèøëû Integrating both sides, we get tanæö=+ç÷èø . 184 MATHEMATICS Case I: dxdy Þ dx =Integrating both sides, we getx= k,Substituting x =1,we getk =Therefore, = 1 is the equation of curve (not possible, so rejected).Case II: dxdy = 2222xydyyxdxxyyxÞ= Substituting y = vx, we get 222dvvxxvxdxvx+= Þ dvvdxv=- = (1)-+ Þ vdxdv Integrating both sides, we getlog (1 +) = ‚ log + loglog (1 +) () = log (1 + =cx. Substituting = 1, = 1,we get = 2.Therefore, =

0is the required equation.Example 9Find the equation of a curve passing through 1,æöç÷èø if the slope of thetangent to the curve at any point P () is cosyyxx SolutionAccording to the given condition cosdyyydxxx=- ... (i)This is a homogeneous differential equation. Substituting =vx, we get dvdx = v €cos dvdx = ‚ cos DIFFERENTIAL EQUATIONS183 Þ yx += e yx k . eÞ kx . eLong Answer (L.A.)Example 8Find the equation of a curve passing through the point (1, 1) if theperpendicular distance of the origin from the normal at any point P() of the curveis equal to the distance of P from the ‚ axis.SolutionLet the equation of normal at P() be Y ‚ = ()X‚dxdy ,i.e.,Y + Xdxdy ‚ dxyxdyæöç÷èø = 0...(1)Therefore, the length of perpendicular from origin to (1) is dxyxdydxdyæöç÷èø ...(2)Also distance between P and-axis is ||. Thus, we get dxyxdydxdyæöÃ

§Ã·Ã¨Ã¸ = |Þ dxyxdyæöç÷èø = dxdyéùæöêúç÷èøêúëû Þ ()22‚20dxdxxyxydydy+= Þ 0dxdy= or dxdy = 22xyyx 182 MATHEMATICS y.x xxdx , i.e. yx = 44xc+ Hencey = xc Example 5Find the differential equation of the family of lines through the origin.SolutionLet =mx be the family of lines through origin. Therefore, dydx Eliminating,we get dydx .x or x dydx ‚ = 0.Example 6Find the differential equation of all non-horizontal lines in a plane.SolutionThe general equation of all non-horizontal lines in a plane is =, whereTherefore, dxabdy = 0.Again, differentiating both sides w.r.t.,we get dxdy = 0 dxdy = 0.Example 7Find the equation of a curve whose tangent at any point on it, differentfrom origin, has slope yyx+ SolutionGiven dyydxx=+ = æöç÷èø Þ dydxyxæö=+ç÷èø Integrating both sides, we getlog + loglog æöç÷èø + DIFFERENTIAL EQUATIONS181 dydx =

2A ‚ 2 B.‚2 and dydx = 4A + 4B‚2Thus dydx = 4i.e., dydx y =Example 2Find the general solution of the differential equation dydx = yx Solution dydx = yx Þ dy = dx Þ dy = dx log = log + logÞ =cxExample 3Given that dydx ye and = 0,e.Find the value of when = 1.Solution dydx ye Þ dy = edx Þlog +Substituting = 0 and,we get log= + c, i.e., = 0 ( log = 1)Therefore, log eNow,substituting = 1 in the above, we get logy = e yExample 4Solve the differential equation dydx + yx 2.SolutionThe equation is of the type +P= Qdydx ,which is a linear differentialequation.Now I.F. = dx log =x.Therefore, solution of the given differential equation is 180 MATHEMATICS (x) A function F () is said to be a homogeneous function of degree ifF (x, l)= l F () for some non-zero constantl.) A differential equation which can be expressed in the form dydx = F () or dxdy = G (), wh

ere F () and G () are homogeneous functions of degreezero, is called a homogeneous differential equation(xii) To solve a homogeneous differential equation of the type dydx = F (), we makesubstitution =vx and to solve a homogeneous differential equation of the type dxdy = G (), we make substitution =vy (xiii) A differential equation of the form dydx + P = Q, where P and Q are constants orfunctions of only is known as a first order linear differential equation. Solutionof such a differential equation is given by (I.F.) = ()QI.F.dx + C, whereI.F. (Integrating Factor) = dx (xiv) Another form of first order linear differential equation is dxdy + P = Q, where and Q are constants or functions of only. Solution of such a differentialequation is given by (I.F.) = ()Q×I.F.dy + C, where I.F. = dy 9.2 Solved ExamplesShort Answer (S.A.)Example 1Find the differential equation of the family of cur

ves = A + B.‚2Solution = A + B.‚2 9.1 Overview(i)An equation involving derivative (derivatives) of the dependent variable withrespect to independent variable (variables) is called a differential equation.(ii)A differential equation involving derivatives of the dependent variable withrespect to only one independent variable is called an ordinary differentialequation and a differential equation involving derivatives with respect to morethan one independent variables is called a partial differential equation.(iii)Order of a differential equation is the order of the highest order derivativeoccurring in the differential equation.(iv)Degree of a differential equation is defined if it is a polynomial equation in itsderivatives.(v)Degree (when defined) of a differential equation is the highest power (positiveinteger only) of the highest order derivative in it.(vi)A relation between invol

ved variables, which satisfy the given differentialequation is called its solution. The solution which contains as many arbitraryconstants as the order of the differential equation is called the general solutionand the solution free from arbitrary constants is called particular solution.(vii)To form a differential equation from a given function, we differentiate thefunction successively as many times as the number of arbitrary constants in thegiven function and then eliminate the arbitrary constants.(viii) The order of a differential equation representing a family of curves is same asthe number of arbitrary constants present in the equation corresponding to thefamily of curves.(ix) €Variable separable method is used to solve such an equation in which variablescan be separated completely, i.e., terms containingshould remain withandterms containing should remain withdy.ChapterD