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1468:{\displaystyle {\frac {\partial \mu }{\partial y}}M+{\frac {\partial M}{\partial y}}\mu ={\frac {\partial \mu }{\partial x}}N+{\frac {\partial N}{\partial x}}\mu .}
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Ordinary
Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences
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1344:{\textstyle {\frac {\partial \mu M}{\partial y}}={\frac {\partial \mu N}{\partial x}}}
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to multiply the equation by. We'll start with the equation itself.
1877:{\displaystyle \mu (x)=e^{\int {{\frac {M_{y}-N_{x}}{N}}\,dx}}.}
1785:{\displaystyle \mu (y)=e^{-\int {{\frac {M_{y}-N_{x}}{M}}\,dy}}}
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In order to solve the equation, we need to transform it into an
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The solution to such equations came with the invention of the
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1561:{\displaystyle M\mu _{y}-N\mu _{x}+(M_{y}-N_{x})\mu =0.}
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a guide for non-partial inexact differential equations
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may be too technical for most readers to understand
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2399:List of nonlinear ordinary differential equations
2404:List of nonlinear partial differential equations
1967:A solution for an inexact differential equation
1902:"History of differential equations – Hmolpedia"
2394:List of linear ordinary differential equations
1997:
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8:
1931:"Recognizable Exact Differential Equations"
53:Learn how and when to remove these messages
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1929:Tenenbaum, Morris; Pollard, Harry (1963).
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239:, without removing the technical details.
110:Learn how and when to remove this message
1697:first-order linear differential equation
152:of all important aspects of the article.
73:This article includes a list of general
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148:Please consider expanding the lead to
16:Solvable form of differential equation
1675:, in which case we only need to find
1258:{\displaystyle \mu M\,dx+\mu N\,dy=0}
237:make it understandable to non-experts
7:
2389:List of named differential equations
412:List of named differential equations
2314:Method of undetermined coefficients
2095:Dependent and independent variables
485:Dependent and independent variables
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1668:{\displaystyle \mu (x,y)=\mu (y)}
1618:{\displaystyle \mu (x,y)=\mu (x)}
195:and remove advice or instruction.
34:This article has multiple issues.
2211:Carathéodory's existence theorem
620:Carathéodory's existence theorem
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2521:Ordinary differential equations
1701:separable differential equation
140:may be too short to adequately
42:or discuss these issues on the
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150:provide an accessible overview
1:
1573:partial differential equation
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978:inexact differential equation
2039:Notation for differentiation
1941:. New York: Dover. pp.
751:Exponential response formula
497:Coupled / Decoupled
2135:Exact differential equation
1142:exact differential equation
2552:
2445:Józef Maria Hoene-Wroński
2425:Gottfried Wilhelm Leibniz
2216:Cauchy–Kowalevski theorem
1478:After simplifying we get
885:Józef Maria Hoene-Wroński
831:Undetermined coefficients
740:Method of characteristics
625:Cauchy–Kowalevski theorem
2339:Finite difference method
610:Picard–Lindelöf theorem
604:Existence and uniqueness
2536:Mathematical structures
2319:Variation of parameters
2309:Separation of variables
2206:Peano existence theorem
2201:Picard–Lindelöf theorem
2088:Attributes of variables
984:of the form (see also:
836:Variation of parameters
826:Separation of variables
615:Peano existence theorem
94:more precise citations.
2516:Differential equations
2480:Carl David Tolmé Runge
2054:Differential-algebraic
2013:Differential equations
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289:Differential equations
2526:Differential calculus
2465:Augustin-Louis Cauchy
2460:Joseph-Louis Lagrange
2354:Finite element method
2344:Crank–Nicolson method
2278:Numerical integration
2257:Exponential stability
2149:Relation to processes
2034:Differential operator
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694:Exponential stability
557:Relation to processes
2531:Discrete mathematics
2359:Finite volume method
2283:Dirac delta function
2252:Asymptotic stability
2194:Existence/uniqueness
2059:Integro-differential
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193:rewrite this article
2369:Perturbation theory
2349:Runge–Kutta methods
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2262:Rate of convergence
2158:(discrete analogue)
813:Perturbation theory
808:Integral transforms
699:Rate of convergence
565:(discrete analogue)
402:Population dynamics
369:Continuum mechanics
360:Applied mathematics
2490:Sofya Kovalevskaya
2324:Integrating factor
2247:Lyapunov stability
2167:Stochastic partial
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2455:Rudolf Lipschitz
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1906:www.eoht.info
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781:Finite volume
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225:This article
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182:This article
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93:
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71:
62:
61:
56:
54:
47:
46:
41:
40:
35:
30:
21:
20:
2485:Martin Kutta
2440:Émile Picard
2420:Isaac Newton
2334:Euler method
2304:Substitution
1936:
1909:. Retrieved
1905:
1896:
1794:
1570:
1477:
1213:, so we get
1139:
1123:
977:
975:
925:Émile Picard
910:Martin Kutta
900:George Green
860:Isaac Newton
692: /
688: /
508: /
374:Chaos theory
269:
251:
245:October 2016
242:
226:
201:October 2016
198:
191:Please help
183:
158:October 2016
155:
139:
137:lead section
106:
100:October 2016
97:
78:
50:
43:
37:
36:Please help
33:
2242:Phase space
2100:Homogeneous
1978:at SOS math
1285:to satisfy
818:Runge–Kutta
563:Difference
506:Homogeneous
318:Engineering
92:introducing
2505:Categories
2470:John Crank
2299:Inspection
2162:Stochastic
2156:Difference
2130:Autonomous
2074:Non-linear
2064:Fractional
2027:Operations
1911:2016-10-16
1888:References
935:John Crank
736:Inspection
690:Asymptotic
574:Stochastic
493:Autonomous
468:Non-linear
458:Fractional
75:references
39:improve it
2511:Equations
2274:solutions
2232:Wronskian
2187:Solutions
2115:Decoupled
2079:Holonomic
1843:−
1826:∫
1806:μ
1754:−
1737:∫
1734:−
1714:μ
1683:μ
1654:μ
1633:μ
1604:μ
1583:μ
1550:μ
1534:−
1509:μ
1502:−
1493:μ
1460:μ
1451:∂
1443:∂
1425:∂
1420:μ
1417:∂
1408:μ
1399:∂
1391:∂
1373:∂
1368:μ
1365:∂
1351:. We get
1333:∂
1325:μ
1322:∂
1307:∂
1299:μ
1296:∂
1273:μ
1237:μ
1221:μ
1155:μ
1132:in 1739.
1100:∂
1092:∂
1086:≠
1077:∂
1069:∂
676:Wronskian
654:Dirichlet
397:Economics
340:Chemistry
330:Astronomy
142:summarize
45:talk page
2382:Examples
2272:Integral
2044:Ordinary
786:Galerkin
686:Lyapunov
597:Solution
541:Notation
533:Operator
519:Features
438:Ordinary
2110:Coupled
2049:Partial
1695:with a
659:Neumann
443:Partial
351:Geology
346:Biology
335:Physics
231:Please
88:improve
2125:Degree
2069:Linear
1949:
846:People
758:
705:Series
463:Linear
302:Fields
77:, but
2174:Delay
2120:Order
1969:from
1943:80–91
1699:or a
980:is a
746:Euler
664:Robin
586:Delay
528:Order
501:Exact
427:Types
295:Scope
1947:ISBN
853:List
1795:or
1625:or
1128:by
976:An
235:to
2507::
1945:.
1933:.
1904:.
1556:0.
988:)
48:.
2005:e
1998:t
1991:v
1955:.
1914:.
1872:.
1866:x
1863:d
1857:N
1851:x
1847:N
1838:y
1834:M
1822:e
1818:=
1815:)
1812:x
1809:(
1777:y
1774:d
1768:M
1762:x
1758:N
1749:y
1745:M
1730:e
1726:=
1723:)
1720:y
1717:(
1663:)
1660:y
1657:(
1651:=
1648:)
1645:y
1642:,
1639:x
1636:(
1613:)
1610:x
1607:(
1601:=
1598:)
1595:y
1592:,
1589:x
1586:(
1553:=
1547:)
1542:x
1538:N
1529:y
1525:M
1521:(
1518:+
1513:x
1505:N
1497:y
1489:M
1463:.
1454:x
1446:N
1437:+
1434:N
1428:x
1411:=
1402:y
1394:M
1385:+
1382:M
1376:y
1336:x
1328:N
1316:=
1310:y
1302:M
1253:0
1250:=
1247:y
1244:d
1240:N
1234:+
1231:x
1228:d
1224:M
1201:0
1198:=
1195:y
1192:d
1188:N
1185:+
1182:x
1179:d
1175:M
1109:.
1103:x
1095:N
1080:y
1072:M
1058:,
1055:0
1052:=
1049:y
1046:d
1042:)
1039:y
1036:,
1033:x
1030:(
1027:N
1024:+
1021:x
1018:d
1014:)
1011:y
1008:,
1005:x
1002:(
999:M
965:e
958:t
951:v
764:)
760:(
276:)
270:(
258:)
252:(
247:)
243:(
229:.
203:)
199:(
188:.
160:)
156:(
146:.
113:)
107:(
102:)
98:(
84:.
55:)
51:(
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