ベルヌーイの微分方程式の一解法
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 32, No. 1 September, 1981. -IfcJSSticW^^B® (US 2 fflSA) ® 32 ^ % 1 -f HSW 56 ^ 9 fi. A Method for Solution of Bernoulli's Differential Equation. Shigeru TAKEUCHI Physics Laboratory, Asahikawa College, Hokkaido University of Education, Asahikawa 070. ^^ m: ^^^ -^ ^%^~?^c7)-w^ ^m^m^±st-w\^^m^'s.. Abstract We consider an application of Bernoulli's differential equation to physical mathematics, and show a solution of the equation with three initial conditions. Imposing the specified conditions, we show that B's equation is a liner differential equation on one hand and a variables separable differential equation on the other.. Introduction The application of a differential equation to physical mathematics is ordinary us values, one of an independent variable, another of a dependent variable.. In mechanics, the independent variable is ordinary in time, the dependent variable involves the displacement of time. In thermodynamics, we may have two independent variables, the volume and temperature, and one dependent variable, the pressure. With electric currents, we may have the current flowing in some part of the circuit as a. dependent variable, while the electromotive force is applied as an independent variable. When in a vacuum tube we measure plate current as function of grid voltage.. In the electromagnetic theory, the electric or magnetic field strength, the dependent variable is a function of four independent variables, the three coordinates, space, and time.. The relation between independent and dependent variables of Bernoulli's differential equation may have one independent variable of the first degree and three dependent variables of the high degree. By using a direct physical application we would like to solve the above differential. (29).
(3) Shigeru TAKEUCHI. equation.. Method for solution In Bernoulli's differential equation put. -d^-+P(x)y=Q(x)yn, ~dx. (1). in which x is a independent variable of the first degree, P{x), Q{x) and y are dependent variables of n which are integral numbers. Now. put. 2. =yl~n,. (2). and choose n so that the term in n does not equal 0 or 1 for all integral numbers.. Then we differentiate on both sides of (2) with respect to x.. dz _ ^ ^ ^ dy Hence -5:—=(l—w).y-"-:;L-; dx ^ '"'-' dx'. dy 1 .„ dz that is, '' dx \-nJ dx. —=. i. ~. ,. y"—. (3). Substituting (3) into (1) we oboain J—y"-d^P(x)y=Q{x)yn,. 1—n '' dx. HenceTl^-^+JW=1-^ that is, -dz-^-(l-n)P(x)z=(l-n)Q(x). (4) Thus we induce linear differential equation (4). Then on both sides of (4) times e<I-")//'<-v)d-1 el'l-">f"w'":+(l-n)zp(x)e^-"}fl'w'ix=(l-n)Q(x)e1. dZ _(l-,i)fO(.v>dA- I /1 --\»A^'l»(l-")/<'>(-V)d^—('1_^'mCy'>n<I-">^<;l'>dA'. dx. and the expression becomes _ {^<I-,,VP(.V)«.V} =(i_ „) Q(x)e{l-")tftwllx (5). Then we integrate with respect to x on both sides of (5) once, zett-"wwclx=(l-n)fQ(x)e(l-"}ft'wdx+c, (6) where Ci is a constant. Hence z=e{"-wl'wdx{-{n-l)fQ(x}e-{"-wt>wdx+c, (7). Substituting (2) into (7) we obtain (30).
(4) A Method for Solution of Bernoulli's Differential Education 1 l-« y=[e("-IW(A')dA'{-(M-l)/0(;v)e-<"-l)p<-Y)d-v+ci}]1-'. (8). Then by substituting n = 0 into (1) we obtain. -J-+P(x)y=Q(x).. (9). Hence (9) is also a linear differential equation. Similarly on both sides of (9) times ef{rmdx 4{/-eflJwdx+p(x)yeflJwdx=Q(x}efpwdxand also -d,^-{yel"wdx}= Q(x)efpwdx. (10). Similarly by integrating with respect to x on both sides of (10) once we obtain yeft'w<lx=fQ{x)efl'wdxdx+C2 where c^ is a constant.. Then simplifying by inserting y into (11) y=e-filw^{fQ(x)e{l'wdxdx+C2}. (12). Similarly by substituting n = 1 into equation number (1) we obtain. ^+P(x)y=Q(x)y.. (13). Simplifying by inserting y and x into (13) dy dx. (14). y ~ Q(x)-P(x)'. then by integrating with respect to y, x on both sides of equation (14) once we obtain. log^f Q(x) P(x)+ca-. (15). dx. Hence y=Ae/<)<-v>-p<-Y' where A=ec', Cs and eca are constant.. By solving the equation described above, we may conclude that the solution of Bernoulli's differential equation is (8), (12) and (15).. References. (1) H. Jeffreys and B. Swirles : Methods of Mathematical Pliysices (Cambrige, 1978) 3rd ed., Chap. 8, p. 244. (2) R. J.Cole : Vector Methods (London, 1972) Chap. 3, p. 40.. (31).
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