Transcription of Differential Equations EXACT EQUATIONS
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Differential Equations EXACT EQUATIONS
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.
(e4x +2xy2)dx+(cosy +2x2y)dy = 0 Exercise 7. (3x2 +ycosx)dx+(sinx−4y3)dy = 0 Theory Answers Integrals Tips Toc JJ II J I Back. Section 2: Exercises 6 Exercise 8. xtan−1 y ·dx+ x2 2(1+y2) ·dy = 0 Exercise 9. (2x+x2y3)dx+(x3y2 +4y3)dy = 0 Exercise 10. (2x3 −3x2y +y3) dy dx
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