Chapter 2: Solution of First order ODE

Transcription

1 0 Chapter : Solution of irst order ODE Se. Separable Equations The differential equation of the form that is is alled separable if f = h g; In order to solve it perform the following steps: Rewrite the equation S as and then integrate to obtain Eample: ind all solutions to d dt Solution: irst we look for the onstant solutions that is we look for the roots of We proeed b separating the two variables to get d dt Then we integrate d dt

2 00 Sine d we get Therefore we have It is not eas to obtain as a funtion of t meaning finding in an epliit form. inall beause there are no onstant solutions all the solutions are given b the impliit equation Homework: Solve the initial value problem Eerises Se. : Solve the following ODEs t Answer: tan t #.. d d e

3 0 #..6 osh6 3sinh d 3sinh d #..8 d d os se tan d d #.. #..0 d d e ln d e #..6 Solve the initial value problem 0 d e e

4 Se.: Homogeneous Equations:Transformation to separable DE 0 d Definition: If the right hand side of the equation f an be epressed as a d funtion of the ratio or alone then we sa the equation is homogenous. Eamples of homogeneous equations d d 0 d d 0 Homogeneous equation d G d 3 d se / d 4 d ln ln d Eamples: the following are OT homogeneous equations d d 0 d d 0 ote: Another wa to determine homogeneit is to verif that eah term has the same degree Solution proedure: Consider the homogeneous equation d f d d where f G G d Let d dv v v v d d so equation beomes dv v Gv whih is a separable DE. dv d d G v v Whih an be solved for v and then replae v b /. Conlusion: Ever homogeneous differential equation an be onverted into a separable differential b the substitution v or v

5 0 Eerises Se. 3 3 # d d 0 # 4 d 3 d

6 0 Se.3: Eat differential equations onsider the funtion given b 4 3 whih is of the form the derivative of the impliit funtion given in is given b d d 3 whih an be written as 0 3 d d whih is of the form 0 d d equation is alled eat DE beause it defines eatl the Total of the two partial derivatives and of the same funtion from equation we an hek this eatness b using the mied partial derivatives as follows 4 4 rom alulus: observe that: i derivatives are equal!

7 Definition: Eat DE A differential equation in the form d d 0 3 is eat if there eists a funtion suh that and 0 so d d d d d Test for eatness: If and are ontinuous funtions and have ontinuous first partial derivatives on some retangle of the -plane then 3 Is eat if and onl if Solution of eat equations: To find the solution of d d 0 whih is we using the following steps: Chek that the equation is indeed eat; Integrate either with respet of the variable or with respet of the variable. The hoie of the equation to be integrated will depend on how eas the alulations are. Let us assume that the first equation was hosen then we get d g The funtion g should be there sine in our integration we assumed that the variable is onstant. 3 Use to find g we have d g' g ' d

8 0 ote that g is a funtion of onl. Therefore in the epression giving g' the variable should disappear. Otherwise something went wrong! 5 Integrate to find g ; 6 Write down the funtion 7 All the solutions are given b the impliit equation 8 If ou are given an IVP plug in the initial ondition to find the onstant. You ma ask what do we do if the equation is not eat? In this ase one an tr to find an integrating fator whih makes the given differential equation eat.this is se.4 Eerise #.3. :Show whether the differential equation are eat or not d d 0

9 0 Eerise #.3.6 Solve 0 d d Chek Eatness Eat is DE So to find integrate Or Integrate : Instead Integrate : g d d To find g To find g g g g 0 ' ' g g g 0 ' ' So So And solution is written as And solution is written as g d d

10 0.3.3 Orthogonal Trajetories Perpendiular urves The meaning of the Orthogonal Trajetorie and the general tehnique to find them will be illustrated b solving the following eamples Eample : Given ind a famil of urves that represent the orthogonal trajetories for these given urves. Solution:. We find the DE of the given urves b eliminating the onstant. ' 3 0 d d. Eah member of the famil of orthogonal trajetories is perpendiular to eah interseting member of the given famil of urves. Therefore the DE desribing the orthogonal trajetories is d d 3. Solve the DE in to get the equation of the orthogonal trajetories d d C and this famil of ellipses should be orthogonal to the original famil of parabolas. These families of urves are plotted in ig.. for positive and. ig.. Orthogonal trajetories for famil of ellipses and parabolas.

11 Eample : ind equations for the orthogonal trajetories for the set of urves ; sketh a few of the given urves and the orthogonal trajetories.. ind De of the given urvesremember to remove the onstant. So the DE of the Orthogonal trajetories is given b - the reiproal of the DE in. 3. To find the orthogonal trajetories we solve this DE in

12 Se.4: Integrating ator Tehnique Assume that the equation d d 0 0 is not eat that is In this ase we look for a funtion whih makes the new equation d d 0 an eat one. The funtion if it eists is alled the integrating fator. ote that sine that equation is eat like we said then we have This is not an ordinar differential equation sine it involves more than one variable. This is what's alled a partial differential equation. These tpes of equations are ver diffiult to solve whih eplains wh the determination of the integrating fator is etremel diffiult eept for the following two speial ases: Case : If the integrating fator is funtion of OLY i.e equation will be of the form d d whih is a separable ordinar differential equation in In this ase the funtion is given b ep d This happens if the epression is a funtion of onl that is the variable disappears from the epression. Case : If the integrating fator is funtion of OLY i.e equation will be of the form d d

13 whih is a separable ordinar differential equation in In this ase the funtion is given b ep d This happens if the epression is a funtion of onl that is the variable disappears from the epression. One the integrating fator is found multipl the old equation b to get a new one whih is eat. Then ou are left to use the previous tehnique of se.3 to solve the new equation. Advie: if ou are not pressured b time hek that the new equation is in fat eat after multipling b the integrating fator! Solution proedure: Let us summarize the above tehnique. Consider the equation If our equation is not given in this form ou should rewrite it first. Step : Chek for eatness that is ompute then ompare them. if it is eat then start solving using previous proedure. Step : Assume that the equation is not eat then evaluate If this epression is a funtion of onl find the integrating fator : a ep d then go to step 4 and 5. Otherwise b evaluate

14 If this epression is a funtion of onl find the integrating fator : then go to step 4 and 5. ep d Otherwise ou an not solve the equation using the tehnique developed above! Step 4: ultipl the old equation b and if ou an hek that ou have a new equation whih is eat Step 5: Solve the new equation using the steps desribed in the previous setionse.3 The following eamples illustrates the use of the integrating fator tehnique: Eerise #.4. ind an integrating fator and solve e d e e d 0

15 Se.5 irst Order Linear Equations A first order linear differential equation has the following form: a d a b * d 0 Divide b a the equation we get is the linear first order differential in standard form d d P q Theorem : If P and q are ontinuous then the differential equation d d P q has the one-parameter famil of solutions General solution q d C ** where e P d is alled the integrating fator. If an initial ondition is given use it to find the onstant C. Proof:

16 SOLUTIO PROCEDURES:. If the differential equation is given as rewrite it in the standard form a d a b * d 0 d d P q P d. ind the integrating fator e. 3. Write down the general solution substitute for q in the general solution formula ** q d C 4. If ou are given an IVP use the initial ondition to find the onstant C.. Eample: ind the partiular solution of: Ans: sin os

17 Eerise.5.6 Solve the equation ' q 0 Where 0 q 3 0 Eerise.5 7 a onsider the differential equation g' ' p g q Where p and q are ontinuous funtions on some interval a b. Assume that g eists. ind a substitution whih redues * to a linear equation and solve b Use the result of a to solve the following equation se d tan d