Differential Equations EXACT EQUATIONS

1 Differential EquationsEXACT EQUATIONSG raham S McDonaldA Tutorial Module for learning the techniqueof solving EXACT differential equationslTable of contentslBegin Tutorialc of on using solutionsFull worked solutionsSection 1: Theory31. TheoryWe consider here the following standard form of ordinary differentialequation ( ):P(x,y)dx+Q(x,y)dy= 0If P y= Q xthen the is said to means that a functionu(x,y) exists such that:du= u xdx+ u ydy=Pdx+Qdy= solves u x=Pand u y=Qto findu(x,y).Thendu= 0 givesu(x,y) =C, whereCis a last equation gives the general solution ofPdx+Qdy= 2: Exercises42. ExercisesClick onExerciselinks for full worked solutions (there are 11exercises in total)Show that each of the following differential EQUATIONS is EXACT anduse that property to find the general solution:Exercise yx2dx= 0 Exercise +y2 2x= 0 Exercise (y+ 1)exdx+ 2(ex 2y)dy= 0lTheorylAnswerslIntegralslTipsTocJJIIJI BackSection 2: Exercises5 Exercise 4.

2 (2xy+ 6x)dx+ (x2+ 4y3)dy= 0 Exercise 5.(8y x2y)dydx+x xy2= 0 Exercise 6.(e4x+ 2xy2)dx+ ( cosy + 2x2y)dy= 0 Exercise 7.(3x2+ycosx)dx+ (sinx 4y3)dy= 0lTheorylAnswerslIntegralslTipsTocJJIIJI BackSection 2: Exercises6 Exercise 1y dx+x22(1 +y2) dy= 0 Exercise 9.(2x+x2y3)dx+ (x3y2+ 4y3)dy= 0 Exercise 10.(2x3 3x2y+y3)dydx= 2x3 6x2y+ 3xy2 Exercise 11.(y2cosx sinx)dx+ (2ysinx+ 2)dy= 0lTheorylAnswerslIntegralslTipsTocJJIIJI BackSection 3: Answers73. , x2=A,3.(y+ 1)ex y2=A, + 3x2+y4=A, (1 y2) + 4y2=A, +x2y2+ siny=A, +ysinx y4=A, 1y=A, +x3y33+y4=A,TocJJIIJIBackSection 3: 2x3y+32x2y2 y44=A, + cosx+ 2y= 4: Standard integrals94. Standard integralsf(x) f(x)dxf(x) f(x)dxxnxn+1n+1(n6= 1)[g(x)]ng (x)[g(x)]n+1n+1(n6= 1)1xln|x|g (x)g(x)ln|g(x)|exexaxaxlna(a>0)sinx cosxsinhxcoshxcosxsinxcoshxsinhxtanx ln|cosx|tanhxln coshxcosecxln tanx2 cosechxln tanhx2 secxln|secx+ tanx|sechx2 tan 1exsec2xtanxsech2xtanhxcotxln|sinx|cothx ln|sinhx|sin2xx2 sin 2x4sinh2xsinh 2x4 x2cos2xx2+sin 2x4cosh2xsinh 2x4+x2 TocJJIIJIBackSection 4: Standard integrals10f(x) f(x)dxf(x) f(x)dx1a2+x21atan 1xa1a2 x212aln a+xa x (0<|x|<a)(a>0)1x2 a212aln x ax+a (|x|>a>0)1 a2 x2sin 1xa1 a2+x2ln x+ a2+x2a (a>0)( a<x<a)1 x2 a2ln x+ x2 a2a (x>a>0) a2 x2a22[sin 1(xa) a2+x2a22[sinh 1(xa)+x a2+x2a2]+x a2 x2a2] x2 a2a22[ cosh 1(xa)+x x2 a2a2]TocJJIIJIBackSection 5.

3 Tips on using solutions115. Tips on using solutionslWhen looking at the THEORY, ANSWERS, INTEGRALS orTIPS pages, use theBackbutton (at the bottom of the page) toreturn to the the solutions intelligently. For example, they can help you getstarted on an exercise, or they can allow you to check whether yourintermediate results are to make less use of the full solutions as you work your waythrough the to exercises12 Full worked solutionsExercise form:P(x,y)dx+Q(x,y)dy= (x,y) = yx2andQ(x,y) =1xEquation is EXACT if P y= Q xCheck: P y= 1x2= Q x is equation EXACT ,u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0and equation has solutionu=C,C= to exercises13 u x=Pgives i) u x= yx2 u y=Qgives ii) u y=1xIntegratei) partially with respect tox,u=yx+ (y),where (y) is an arbitrary function respect toy, u y=1x+ y=1x+d dy(since = (y) only)TocJJIIJIBackSolutions to exercises14 Compare with equation ii)1x+d dy= dy= =C ,C = constantandu=yx+C.

4 Du= 0 impliesu=C,C= constant yx=A,A=C C = to Exercise 1 TocJJIIJIBackSolutions to exercises15 Exercise form: (y2 2x)dx+ 2xydy= 0 EXACT if P y= Q x,whereP(x,y) =y2 2xQ(x,y) = 2xy P y= 2y= Q is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,givingi) u x=y2 2x,ii) u y= i):u=xy2 x2+ (y), is arbitrary compare with ii): u y= 2xy+d dy= 2xyTocJJIIJIBackSolutions to exercises16 d dy= 0 and =C (constant) u=xy2 x2+C du= 0 impliesu=C, xy2 x2=A,whereA=C C .Return to Exercise 2 TocJJIIJIBackSolutions to exercises17 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) = 2(y+ 1)exQ(x,y) = 2(ex 2y) P y= 2ex= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x= 2(y+ 1)ex,ii) u y= 2(ex 2y) .Integrate i):u= 2(y+ 1)ex+ (y)Differentiate: u y= 2ex+d dy= 2(ex 2y) , using ii)TocJJIIJIBackSolutions to dy= d = 4 = 2y2+C u= 2(y+ 1)ex 2y2+C du= 0 givesu=C, (y+ 1)ex y2=A, whereA= (C C )/2.

5 Return to Exercise 3 TocJJIIJIBackSolutions to exercises19 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) = 2xy+ 6x,Q(x,y) =x2+ 4y3 P y= 2x= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x= 2xy+ 6x,ii) u y=x2+ i):u=x2y+ 3x2+ (y)Differentiate: u y=x2+d dy=x2+ 4y3, using ii) dy= d = 4 =y4+C TocJJIIJIBackSolutions to exercises20 u=x2y+ 3x2+y4+C du= 0 givesu=C, x2y+ 3x2+y4=A, whereA=C C .Return to Exercise 4 TocJJIIJIBackSolutions to exercises21 Exercise 5.(x xy2)dx+ (8y x2y)dy= 0P(x,y) =x xy2Q(x,y) = 8y x2y. P y= 2xy= Q x, is EXACT . u(x,y) exists wheredu= u xdx+ u ydy=Pdx+Qdy= 0 Givingi) u x=x xy2;ii) u y= 8y i):u=12x2(1 y2) + (y)Differentiate: u y= 12x2 2y+d dy= 8y x2y, using ii)TocJJIIJIBackSolutions to exercises22 d dy= d = 8 (y) = 4y2+C andu=12x2(1 y2) + 4y2+C du= 0 givesu=C, 12x2(1 y2) + 4y2=A,A=C C.

6 Return to Exercise 5 TocJJIIJIBackSolutions to exercises23 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) =e4x+ 2xy2,Q(x,y) = cosy + 2x2y P y= 4xy= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x=e4x+ 2xy2,ii) u y= cosy + i):u=14e4x+x2y2+ (y)Differentiate: u y= 2x2y+d dy= cosy + 2x2y, using ii) dy= d = = siny+C TocJJIIJIBackSolutions to exercises24 u=14e4x+x2y2+ siny+C du= 0 givesu=C, 14e4x+x2y2+ siny=A, whereA=C C .Return to Exercise 6 TocJJIIJIBackSolutions to exercises25 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) = 3x2+ycosxQ(x,y) = sinx 4y3 P y= cosx= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0 Givingi) u x= 3x2+ycosx,ii) u y= sinx i):u=x3+ysinx+ (y)Differentiate: u y= sinx+d dy= sinx 4y3, using ii)TocJJIIJIBackSolutions to exercises26 d dy= d = 4 = y4+C andu=x3+ysinx y4+C du= 0 givesu=C, x3+ysinx y4=A,A=C C.

7 Return to Exercise 7 TocJJIIJIBackSolutions to exercises27 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) =xtan 1yQ(x,y) =x22(1+y2) P y=x1+y2= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x=xtan 1y,ii) u y=x22(1+y2).Integrate i):u=x22tan 1y+ (y)Differentiate: u y=x221(1+y2)+d dy=x22(1+y2), using ii)TocJJIIJIBackSolutions to exercises28 d dy= (y) =C andu=x22tan 1y+C du= 0 impliesu=C,C= constant x22tan 1y=A,A=C C .Return to Exercise 8 TocJJIIJIBackSolutions to exercises29 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) = 2x+x2y3,Q(x,y) =x3y2+ 4y3 P y= 3x2y2= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x= 2x+x2y3,ii) u y=x3y2+ i):u=x2+x3y33+ (y)Differentiate: u y=x3y2+d dy=x3y2+ 4y3, using ii) dy= d = 4 =y4+C TocJJIIJIBackSolutions to exercises30 u=x2+x3y33+y4+C du= 0 givesu=C, x2+x3y33+y4=A, whereA=C C.

8 Return to Exercise 9 TocJJIIJIBackSolutions to exercises31 Exercise 10.(2x3 6x2y+ 3xy2)dx+ ( 2x3+ 3x2y y3)dy= 0 P y= 6x2+ 6xy= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0 Giving i) u x= 2x3 6x2y+ 3xy2, ii) u y= 2x3+ 3x2y i):u=x42 2x3y+32x2y2+ (y)Differentiate: u y= 2x3+ 3x2y+d dy= 2x3+ 3x2y y3, using ii)TocJJIIJIBackSolutions to exercises32 d dy= d = (y) = 14y4+C andu(x,y) =x42 2x3y+32x2y2 y44+C du= 0 givesu=C, x42 2x3y+32x2y2 y44=A,A=C C .Return to Exercise 10 TocJJIIJIBackSolutions to exercises33 Exercise (x,y)dx+Q(x,y)dy= 0 whereP(x,y) =y2cosx sinx,Q(x,y) = 2ysinx+ 2 P y= 2ycosx= Q x, is EXACT . u(x,y) exists such thatdu= u xdx+ u ydy=Pdx+Qdy= 0,Givingi) u x=y2cosx sinx,ii) u y= 2ysinx+ 2 .Integrate i):u=y2sinx+ cosx+ (y)Differentiate: u y= 2ysinx+d dy= 2ysinx+ 2 , using ii) dy= d = 2 = 2y+C TocJJIIJIBackSolutions to exercises34 u=y2sinx+ cosx+ 2y+C du= 0 givesu=C, y2sinx+ cosx+ 2y=A, whereA=C C.

9 Return to Exercise 11 TocJJIIJIBack