Mathematics
Separable Equations
Separable equations in mathematics refer to differential equations that can be written in a form where the variables can be separated and then integrated individually. This allows for the solution of the differential equation by integrating both sides. The process involves isolating the variables and then integrating with respect to each variable separately.
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10 Key excerpts on "Separable Equations"
eBook - PDF- Dennis Zill(Author)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 2.2 Separable Equations 47 INTRODUCTION We begin our study of solution methods with the simplest of all differential equations: first-order equations with separable variables. Because the method discussed in this section and many other methods for solving differential equations involve integration, you are urged to refresh your memory on important formulas (such as e du y u ) and techniques (such as integration by parts) by consulting a calculus text. SOLUTION BY INTEGRATION Consider the first-order differential equation dy y dx 5 f ( x , y ). When f does not depend on the variable y , that is, f ( x , y ) 5 g ( x ), the differential equation dy dx 5 g ( x ) (1) can be solved by integration. If g ( x ) is a continuous function, then integrating both sides of (1) gives y 5 e g ( x ) dx 5 G ( x ) 1 c , where G ( x ) is an antiderivative (indefinite integral) of g ( x ). For example, if dy y dx 5 1 1 e 2 x , then its solution is y 5 e (1 1 e 2 x ) dx or y 5 x 1 1 2 e 2 x 1 c . A DEFINITION Equation (1), as well as its method of solution, is just a special case when the function f in the normal form dy y dx 5 f ( x , y ) can be factored into a function of x times a function of y. 2.2 Separable Equations For example, the equations dy dx 5 y 2 xe 3 x 1 4 y and dy dx 5 y 1 sin x (c) Verify that an explicit solution of the DE in the case when k 5 1 and a 5 b is X ( t ) 5 a 2 1 y ( t 1 c ). Find a solution that satisfies X (0) 5 a y 2. Then find a solution that satisfies X (0) 5 2 a . Graph these two solutions. Does the behavior of the solutions as t : ` agree with your answers to part (b)? where k . 0 is a constant of proportionality and b . a . 0. Here X ( t ) denotes the number of grams of the new compound formed in time t. (a) Use a phase portrait of the differential equation to predict the behavior of X ( t ) as t : ` .
eBook - PDF- Martha L Abell, James P. Braselton(Authors)
- 2014(Publication Date)
- Academic Press(Publisher)
FirsuOrder Ordinary Differential Equations In this chapter we introduce frequently used first-order ordinary differential equations and methods to construct their solutions. The equations and methods of solution found in this chapter are standard. Although much of the material in this chapter is briefly discussed, several of the equations found here will be used in other chapters of the text. 2. 1 S E P A R A T I O N O F V A R I A B L E S Differential equations are first encountered in beginning integral calculus courses. Although the phrase differential equation is not frequently used at that point, the problem of finding a function whose derivative is a given function is a differential equation. D E F I N I T I O N Separable Differential Equation A differential equation that can be written in the form giy)y' = m is called a separable differential equation. Separable differential equations are solved by collecting all the terms involving y on one side of the equation and all the terms involving χ on the other side of the equation and integrating. Rewriting g{y)y' = f(x) in the form 18 Differential Equations with Maple V yields g(i/) dy = f{x) dx so that giy)dy = f{x)dx --C, Ca constant. Therefore, in the case of a separable differential equation, we simply separate the variables and integrate both sides of the equation. EXAMPLE: Show that the equation dy ^ ly ^f^ -2y dx X is separable, and solve by separation of variables. dy 2y^^^ - 2y SOLUTION: The equation ^ = —— -- is separable since it can be written in the form dx 2yi /2 -2y x' To solve the equation, integrate both sides and simplify. Observe that 2yi /2 -2y is the same as dV 2yi /2 (1 - yi /2) X = — ^ Wethenobtain To evaluate the integral on the left-hand side, let w = 1 - y^/^ so that du = 2yU2' = f ^ + Ci so that - nu = nx + Ci. Recall that - nu = n-. Then - = Cx, where U J X u u C = and resubstituting we find that 1 - yl /2 = Cx dy 2y ^/2 - 2y is a general solution of the equation — = --.
eBook - PDFDifferential Equations
An Introduction to Modern Methods and Applications
- James R. Brannan, William E. Boyce(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
We will now show that this process is actually applicable to a much larger class of equations. We will use x, rather than t, to denote the independent variable in this section for two rea- sons. In the first place, different letters are frequently used for the variables in a differential equation, and you should not become too accustomed to using a single pair. In particular, x often occurs as the independent variable. Further, we want to reserve t for another purpose later in this section. The general first order equation is dy dx = f (x, y). (3) The equations we want to consider first are called separable, because the right side f (x, y) has a special form. DEFINITION 2.1.1 Separable Differential Equation. If the right side f (x, y) of Eq. (3) can be written as the product of a function that depends only on x times another function that depends only on y, dy dx = f (x, y) = p(x)q(y), (4) then the equation is called separable. If a differential equation is separable, Definition 2.1.1 means we can find two such functions p and q. For example, Eq. (2) written in the variables (x, y) becomes dy dx = f (x, y) = ay + b. (5) One choice for the functions p and q for Eq. (5) is p(x) = 1 and q(y) = ay + b, and therefore Eq. (5) is separable. Once we know a differential equation is separable, we can find an expression for the solution by integration. A convenient shortcut uses the differentials dx and dy = y ′ (x) dx, and multiplying Eq. (4) by dx gives dy = p(x)q(y) dx. (6) We assume q(y) is nonzero for y values of interest, divide Eq. (6) by q, and integrate both sides to produce ∫ q(y) −1 dy = ∫ p(x) dx. (7) Substituting our choices for p and q, we can integrate both sides of Eq. (7). For example, to solve Eq. (5), the same calculation steps can be used as in Section 1.1 [where the solution Eq. (8) is found from Eq. (2) in that section], y(x) = − b a + ce ax (c ≠ 0). (8)
- Victor Yu Reshetnyak, Mikola Vladimirovich Makarets(Authors)
- 1995(Publication Date)
- World Scientific(Publisher)
C h a p t e r 1 F I R S T O R D E R D I F F E R E N T I A L E Q U A T I O N S 1.1 Separable Equations A differential equation which can be written in the form M(x)dx + N(y)dy = 0, (1) where M is a function of X alone and N is a function of y alone, is said to be separable. The solution is j M{x)dx + j N(y)dy = C, (2) where C is an arbitrary constant. The problem is then reduced to the problem of evaluating the two integrals in (2). In Eq.(l) we say that the variables are separated. Example 1. Find the solution of the equation y' = e'*> which is such that y — 0 when x = 0. The equation may be written as y' = eV, from which it is seen that the separated form is e~*dy — e'dx. Integrating now gives the general solution -e~> = t z + C, and we have to find the value of the constant C such that x and y vanish simultane-ously. On putting t = y — 0, we have — 1 = 1 + C whence C = —2. The appropriate solution is given by e* = 2 -e' 1 2 CHAPTER 1. FIRST ORDER DIFFERENTIAL EQUATIONS Example 2. Solve the equation xydx+(x+l)dy = 0. (3) If y j£ 0 and x + 1 ^ 0, we can divide by tj and i + I and put the equation in the form dy xdx Integrating, = 0. J y J x + 1 Mjrl + i -l n x + l = C. Taking exponential of both sides yields j z -d ^ + lje 1 , C, = In |C|. Equation (3) has also solutions y = 0 and x = — 1 The first one can be obtained from the general solution when arbitrary constant C — 0 and therefore JJ = 0 is the particular solution. The second solution x — — 1 can't be obtained from the genera! solution and therefore x = — 1 is the singular solution. Then the solution of the problem (3) is y = C,(x+ )e-* if x jt -1; also I = -1 . Example 3. Solve the initial value problem y , coti + y = 2; y ( j ] = 0. (4) KJf 2 and cot x ^ 0 the differential equation can be written as dy Integrating, Whence • + tan xdx — 0. f dy l sin xdx _ ^ I y — 2 J cos x ' l n | s -2 | -l n | e o s i | = C. B = 2+&eosjf ) (5) where C, = ]aG is an arbitrary constant.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
2.2 Separable Differential Equations 35 The general first-order differential equation is = (, ). (2) Linear differential equations were considered in the preceding section, but if equation (2) is nonlinear, then there is no universally applicable method for solving the equation. Here, we consider a subclass of first-order equations that can be solved by direct integration. To identify this class of equations, we first rewrite equation (2) in the form (, ) + (, ) = 0. (3) It is always possible to do this by setting (, ) = −(, ) and (, ) = 1, but there may be other ways as well. When is a function of only and is a function of only, then equation (3) becomes () + () = 0. (4) Such an equation is said to be separable, because if it is written in the differential form () + () = 0, (5) then, if you wish, terms involving each variable may be placed on opposite sides of the equation. The differential form (5) is also more symmetric and tends to suppress the distinction between independent and dependent variables. A separable equation can be solved by integrating the functions and . We illustrate the process by an example and then discuss it in general for equation (4). EXAMPLE 2.2.1 Show that the equation = 2 1 − 2 (6) is separable, and then find an equation for its integral curves. Solution If we write equation (6) as − 2 + ( 1 − 2 ) = 0, (7) then it has the form (4) and is therefore separable. Recall from calculus that if is a function of , then by the chain rule, () = () = ′ () . For example, if () = − 3 ∕3, then ( − 3 3 ) = ( 1 − 2 ) . Thus the second term in equation (7) is the derivative with respect to of − 3 ∕3, and the first term is the derivative of − 3 ∕3. Thus equation (7) can be written as (− 3 3 ) + ( − 3 3 ) = 0, or (− 3 3 + − 3 3 ) = 0. ▼ ▼
eBook - PDF- Wei-Chau Xie(Author)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
2 C H A P T E R First-Order and Simple Higher-Order Differential Equations There are various techniques for solving first-order and simple higher-order ordi-nary differential equations. The key in the application of the specific technique hinges on the identification of the type of a given equation. The objectives of this chapter are to introduce various types of first-order and simple higher-order dif-ferential equations and the corresponding techniques for solving these differential equations. In this chapter, it is assumed that x is the independent variable and y is the dependent variable. Solutions in the explicit form y = η( x ) or in the implicit form u ( x , y ) = 0 are sought. 2.1 The Method of Separation of Variables Consider a first-order ordinary differential equation of the form d y d x = F ( x , y ). Suppose that the right-hand side F ( x , y ) , which is a function of x and y , can be written as a product of a function of x and a function of y , i.e., F ( x , y ) = f ( x ) · φ( y ). For example, the functions e x + y 2 = e x · e y 2 , xy + x + 2 y + 2 = ( x + 2 ) · ( y + 1 ) 16 2.1 the method of separation of variables 17 can be separated into a product of a function of x and a function of y , but the following functions cannot be separated ln ( x + 2 y ) , sin ( x 2 + y ) , xy 2 + x 2 . This type of differential equation is called variable separable or separable differential equations. The equations can be solved by the method of separation of variables. Rewrite the equation as d y d x = f ( x ) · φ( y ). Case 1. If φ( y ) = 0, moving terms involving variable y to the left-hand side and terms of variable x to the right-hand side yields g ( y ) d y = f ( x ) d x , g ( y ) = 1 φ( y ) . function of y only function of x only Integrating both sides of the equation results in the general solution g ( y ) d y = f ( x ) d x + C , where C is an arbitrary constant.
eBook - PDFCalculus
One and Several Variables
- Saturnino L. Salas, Garret J. Etgen, Einar Hille(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
CHAPTER 9 SOME DIFFERENTIAL EQUATIONS Introduction An equation that relates an unknown function to one or more of its derivatives is called a differential equation. We have already introduced some differential equations. In Chapter 7 we used the differential equation (1) dy dt = ky [there written f (t) = k f (t)] to model exponential growth and decay. In various exercises (Section 3.6 and 4.9) we used the differential equation (2) d 2 y dt 2 + ω 2 y = 0, the equation of simple harmonic motion, to model the motion of a simple pendulum and the oscillation of a weight suspended at the end of a spring. The order of a differential equation is the order of the highest derivative that appears in the equation. Thus (1) is a first-order equation and (2) is a second-order equation. A function that satisfies a differential equation is called a solution of the equation. Finding the solutions of a differential equation is called solving the equation. All functions y = Ce kt where C is a constant are solutions of equation (1): dy dt = kCe kt = ky . All functions of the form y = C 1 cos ωt + C 2 sin ωt , where C 1 and C 2 constants are solutions of equation (2): y = C 1 cos ωt + C 2 sin ωt dy dt = −ωC 1 sin ωt + ωC 2 cos ωt d 2 y dt 2 = −ω 2 C 1 cos ωt − ω 2 C 2 sin ωt = −ω 2 y 443 444 ■ CHAPTER 9 SOME DIFFERENTIAL EQUATIONS and therefore d 2 y dt 2 + ω 2 y = 0. Remark Differential equations reach far beyond the boundaries of pure mathematics. Countless processes in the physical sciences, in the life sciences, in engineering, and in the social sciences are modeled by differential equations. The study of differential equations is a huge subject, certainly beyond the scope of this text or any text on calculus. In this little chapter we examine some simple, but useful, differential equations. We continue the study of differential equations in Chapter 19.
- Barbara D. MacCluer, Paul S. Bourdon, Thomas L. Kriete(Authors)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
Chapter 2 First-Order Equations 2.1. Linear equations The goal of this section and the next is to develop two basic methods for finding solutions to certain types of first-order differential equations. After the classification of a differential equation as either an ordinary differential equation or a partial differential equation, and after deciding on the order of the equation, the next crucial step is to classify the equation as linear or nonlinear. A first-order ordinary differential equation is linear if it can be written in the form (2.1) dy dt + p ( t ) y = g ( t ); we refer to this as the standard form of a first-order linear equation. Here t is the independent variable (we use the letter t frequently because in applications it often plays the role of time) and y is the dependent variable. If we compare this standard form for a first-order linear equation with the normal form dy dt = f ( t,y ) of any first-order equation, we see that f ( t,y ) = − p ( t ) y + g ( t ) , so that f ( t,y ) is a linear function of the variable y . The coefficient functions p ( t ) and g ( t ) are usually assumed to be continuous functions, at least on some open interval a < t < b . If the function g ( t ) happens to be the constant 0, equation (2.1) is said to be homogeneous . A first-order equation that cannot be written in the form (2.1) is called nonlinear . For example, the equation dy dt = cos t + 3 y is linear (with p ( t ) = − 3 and g ( t ) = cos t when we put the equation in standard form), while the equation dy dt = cos y + 3 t is nonlinear. The equation 2 ty + t cos t + ( t 2 + 1) dy dt = 0 is also linear, since upon dividing by t 2 + 1 it can be rewritten as dy dt + 2 t t 2 + 1 y = − t cos t t 2 + 1 , 25 26 2. First-Order Equations and we see that it has the form of equation (2.1) with p ( t ) = 2 t t 2 + 1 and g ( t ) = − t cos t t 2 + 1 .
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Each of these statements involves a rate of 40 CHAPTER 2 First-Order Differential Equations change (derivative) and consequently, when expressed mathematically, leads to a differential equation. The differential equation is a mathematical model of the process. It is important to realize that the mathematical equations are almost always only an approximate description of the actual process. For example, bodies moving at speeds comparable to the speed of light are not governed by Newton’s laws, insect populations do not grow indefinitely as stated because of eventual lack of food or space, and heat transfer is affected by factors other than the temperature difference. Thus you should always be aware of the limitations of the model so that you will use it only when it is reasonable to believe that it is accurate. Alternatively, you can adopt the point of view that the mathematical equations exactly describe the operation of a simplified physical model, which has been constructed (or conceived of) so as to embody the most important features of the actual process. Sometimes, the process of mathematical modeling involves the conceptual replacement of a discrete process by a continuous one. For instance, the number of members in an insect population changes by discrete amounts; however, if the population is large, it seems reasonable to consider it as a continuous variable and even to speak of its derivative. Step 2: Analysis of the Model. Once the problem has been formulated mathematically, you are often faced with the problem of solving one or more differential equations or, failing that, of finding out as much as possible about the properties of the solution. It may happen that this mathematical problem is quite difficult, and if so, further approximations may be indicated at this stage to make the problem mathematically tractable. For example, a nonlinear equation may be approximated by a linear one, or a slowly varying coefficient may be replaced by a constant.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
Determine how the solution behaves as t → ∞ Homogeneous Equations. If the right-hand side of the equation dy / dx = f ( x , y ) can be expressed as a function of the ratio y / x only, then the equation is said to be homogeneous. 1 Such equations can always be transformed into Separable Equations by a change of the dependent variable. Problem 25 illustrates how to solve first-order homogeneous equations. .............................................................................................................................. 1 The word “homogeneous” has different meanings in different mathematical contexts. The homogeneous equations considered here have nothing to do with the homogeneous equations that will occur in Chapter 3 and elsewhere. 2.3 Modeling with First-Order Differential Equations 39 N 25. Consider the equation dy dx = y − 4x x − y . (29) a. Show that equation (29) can be rewritten as dy dx = ( y / x ) − 4 1 − ( y / x ) ; (30) thus equation (29) is homogeneous. b. Introduce a new dependent variable v so that v = y / x , or y = xv ( x ). Express dy / dx in terms of x , v , and dv / dx . c. Replace y and dy / dx in equation (30) by the expressions from part b that involve v and dv / dx . Show that the resulting differential equation is v + x dv dx = v − 4 1 − v , or x dv dx = v 2 − 4 1 − v . (31) Observe that equation (31) is separable. d. Solve equation (31), obtaining v implicitly in terms of x . e. Find the solution of equation (29) by replacing v by y / x in the solution in part d. f. Draw a direction field and some integral curves for equation (29). Recall that the right-hand side of equation (29) actually depends only on the ratio y / x . This means that integral curves have the same slope at all points on any given straight line through the origin, although the slope changes from one line to another. Therefore, the direction field and the integral curves are symmetric with respect to the origin.
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