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In Mathematics , and particularly in Analysis , an ordinary differential equation (or '''ODE''') is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. See Differential Calculus and Integral Calculus for basic calculus background. Many scientific theories can be expressed clearly and concisely in terms of ordinary differential equations. For instance, the law for radioactive decay of a single isotope of an element, states that its rate of loss of mass is proportional to its mass. If ''t'' represents time and ''u(t)'' represents the mass of the isotope at time ''t'', then the law for decay states that : where $\backslash alpha$ is a constant that depends upon the particular isotope. Another example of an ordinary differential equation is Newton's second law of motion of a single particle, which states that ''f = ma'', where ''f'' is an applied force, ''m'' is the mass of the particle, and ''a'' is the acceleration of the particle due to the force. If motion is constrained to a straight line, the coordinate ''t'' measures the time elapsed and the unknown function ''x(t)'' specifies the position of the particle along the line, then the velocity of the particle ''v'' is given by the first derivative of ''x'' with respect to ''t'': :. Similarly, the acceleration of the particle ''a'' is given by the second derivative of ''x'': :. Thus Newton's second law implies the differential equation :. In general, the force depends upon the position of the particle, and thus the unknown function ''x'' appears on both sides of the differential equation, as is indicated in the notation ''f(x)''. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Bendixson Theorem . DEFINITION Let ''y'' represent an unknown function of ''x'', and let :$y\text{'},\; y\text{'}\text{'},\backslash \; \backslash dots,\backslash \; y^\{(n)\}$ denote the Derivative s : An ordinary differential equation (ODE) is an equation involving :$x,\backslash \; y,\backslash \; y\text{'},\backslash \; y\text{'}\text{'},\backslash \; \backslash dots\; .$ The order of a differential equation is the order ''n'' of the highest derivative that appears. If the highest derivative appears only in integer powers, then the '''degree''' of the equation is the highest power of the highest derivative. A solution of an ODE is a function ''y''(''x'') whose derivatives satisfy the equation. Such a function is not guaranteed to exist and, if it does exist, is usually not unique. A '''general solution''' of an ''n''th-order equation is a solution containing $n$ arbitrary variables, corresponding to ''n'' Constants Of Integration . A '''particular solution''' is derived from the general solution by setting the constants to particular values. A Singular Solution is a solution that can't be derived from the general solution. When a differential equation of order ''n'' has the form :$F\backslash left(x,\; y\text{'},\; y\text{'}\text{'},\backslash \; \backslash dots,\backslash \; y^\{(n)\}\; ight)\; =\; 0$ it is called an implicit differential equation whereas the form :$F\backslash left(x,\; y\text{'},\; y\text{'}\text{'},\backslash \; \backslash dots,\backslash \; y^\{(n-1)\}\; ight)\; =\; y^\{(n)\}$ is called an explicit differential equation. A differential equation not depending on ''x'' is called autonomous, and one with no terms depending ''only'' on ''x'' is called '''homogeneous'''. GENERAL APPLICATION An important special case is when the equations do not involve $x$. These differential equations may be represented as Vector Field s. This type of differential equation has the property that space can be divided into Equivalence Class es based on whether two points lie on the same solution Curve . Since the laws of physics are believed not to change with time, the physical world is governed by such differential equations. (See also Symplectic Topology for abstract discussion.) In the case where the equations are Linear , the original equation can be solved by breaking it down into smaller equations, solving those, and then adding the results back together. Unfortunately, many of the interesting differential equations are non-linear, which means that they cannot be broken down in this way. There are also a number of techniques for solving differential equations using a computer (see Numerical Ordinary Differential Equations ). Ordinary differential equations are to be distinguished from Partial Differential Equation s where $y$ is a function of several variables, and the differential equation involves Partial Derivative s. EXISTENCE AND NATURE OF SOLUTIONS The problem of solving a differential equation is to find the function $y$ whose derivatives satisfy the equation. For example, the differential equation :$y\text{'}\text{'}\; +\; y\; =\; 0\; \backslash ,$ has the general solution :$y\; =\; A\; \backslash cos\{x\}\; +\; B\; \backslash sin\{x\}\; \backslash ,$, where ''A'', ''B'' are constants determined from Boundary Condition s. In general, an ''n''-th order equation allows both $x$ and $y$ to be fixed, as well as all the $n-1$ lower order derivatives of $y$; the remaining equation can be solved (at least conceptionally) for $y^\{(n)\}$. If the equation has finite degree $d$, then we now have a polynomial equation in $y^\{(n)\}$ with at most $d$ roots. Therefore there can be as many as $d$ possible values for $y^\{(n)\}$ at any given point and for any possible values of the lower order derivatives, though there may be ranges of these points and values where there are fewer solutions (or none at all). A Lipschitz Condition must also be satisfied for a solution to exist. Thus, in the previous example, a second-order, first-degree equation, any point on the plane and any slope through that point can be selected and yield a unique solution (since the single root of $y\text{'}\text{'}$ exists for any value of $y$). Note in particular that there are an infinity of solutions through any given point; this is a general characteristic of equations of order higher than one. Consider now :$(y\text{'})^2\; +\; xy\text{'}\; -\; y\; =\; 0\; \backslash ,$ with general solution :$y\; =\; Ax\; +\; A^2\; \backslash ,$ This is a first-order, second-degree equation, thus any point can have at most two solutions passing through it, corresponding to the two roots of $y\text{'}$ in the Quadratic Equation that would result after fixing $x$ and $y$. Studying the quadratic equation's Discriminant ($x^2\; +\; 4y$) leads to the conclusion that only a single solution exists along the parabola (where the discriminant is zero) and that no solution exists below this parabola (where both roots are complex). The parabola in this problem is an example of a cusp locus; a curve along which two or more roots of the differential equation are identical. Along such a locus it is possible to move from one general solution to another while still obeying the differential equation; thus the presence of cusp loci introduce the possibility of '''singular''' solutions. In this example, the parabola is such a singular solution; it satisfies the original differential equation, and a full set of solutions must include such possibilities as the hybrid solution: where the cusp locus has been used to connect two particular solutions; note that the first derivative (the only derivative to appear in the differential equation) is continuous at the transitions. ( ''Johnson'' , Chapter 5) TYPES OF DIFFERENTIAL EQUATIONS WITH SOME HISTORY The influence of geometry, physics, and astronomy, starting with Newton and Leibniz , and further manifested through the Bernoullis , Riccati , and Clairaut , but chiefly through D'Alembert and Euler , has been very marked, and especially on the theory of linear partial differential equations with constant coefficients. Homogeneous linear ODEs with constant coefficients The first method of integrating linear ordinary differential equations with constant coefficients is due to Euler , who realized that solutions have the form $e^\{z\; x\}$, for possibly-complex values of $z$. Thus : has the form :$z^n\; e^\{zx\}\; +\; A\_1\; z^\{n-1\}\; e^\{zx\}\; +\; \backslash cdots\; +\; A\_n\; e^\{zx\}\; =\; 0$ so dividing by $e^\{zx\}$ gives the ''n''th-order polynomial :$F(z)\; =\; z^\{n\}\; +\; A\_\{1\}z^\{n-1\}\; +\; \backslash cdots\; +\; A\_n\; =\; 0$ In short the terms : of the original differential equation are replaced by ''z''''k''. Solving the polynomial gives ''n'' values of ''z'', $z\_1,\; \backslash dots,z\_n$. Plugging those values into $e^\{z\_i\; x\}$ gives a Basis for the solution; any Linear Combination of these basis functions will satisfy the differential equation. This equation ''F''(''z'') = 0, is the "characteristic" equation considered later by Monge and Cauchy . If ''z'' is a (possibly not real) Zero of ''F''(''z'') of multiplicity ''m'' and $k\backslash in\backslash \{0,1,\backslash dots,m-1\backslash \}\; \backslash ,$ then $y=x^ke^\{zx\}\; \backslash ,$ is a solution of the ODE. These functions make up a Basis of the ODE's solutions. If the ''Ai'' are real then real-valued solutions are preferable. Since the non-real ''z'' values will come in Conjugate pairs, so will their corresponding ''y''s; replace each pair with their Linear Combination s Re(''y'') and Im(''y'') . A case that involves complex roots can be solved with the aid of Euler's Formula . Example: Given $y\text{'}\text{'}-4y\text{'}+5y=0\; \backslash ,$. The characteristic equation is $z^2-4z+5=0\; \backslash ,$ which has zeroes 2+''i'' and 2−''i''. Thus the solution basis $\backslash \{y\_1,y\_2\backslash \}$ is $\backslash \{e^\{(2+i)x\},e^\{(2-i)x\}\backslash \}\; \backslash ,$. Now ''y'' is a solution Iff $y=c\_1y\_1+c\_2y\_2\; \backslash ,$ for $c\_1,c\_2\backslash in\backslash mathbb\; C$. Because the coefficients are real, we are likely not interested in the complex solutions our basis elements are mutual conjugates The linear combinations : and : will give us a real basis in $\backslash \{u\_1,u\_2\backslash \}$. Linear ODEs with constant coefficients Suppose instead we face : For later convenience, define the characteristic polynomial :$P(v)=v^n+A\_1v^\{n-1\}+\backslash cdots+A\_n.$ We find the solution basis $\backslash \{y\_1,y\_2,\backslash ldots,y\_n\backslash \}$ as in the homogeneous (''f''=0) case. We now seek a particular solution ''yp'' by the '''variation of parameters''' method. Let the coefficients of the linear combination be functions of ''x'': :$y\_p=u\_1y\_1+u\_2y\_2+\backslash cdots+u\_ny\_n.$ Using the "operator" notation $D=d/dx$ and a broad-minded use of notation, the ODE in question is $P(D)y=f$; so :$f=P(D)y\_p=P(D)(u\_1y\_1)+P(D)(u\_2y\_2)+\backslash cdots+P(D)(u\_ny\_n).$ With the constraints :$0=u\text{'}\_1y\_1+u\text{'}\_2y\_2+\backslash cdots+u\text{'}\_ny\_n$ :$0=u\text{'}\_1y\text{'}\_1+u\text{'}\_2y\text{'}\_2+\backslash cdots+u\text{'}\_ny\text{'}\_n$ :… :$0=u\text{'}\_1y^\{(n-2)\}\_1+u\text{'}\_2y^\{(n-2)\}\_2+\backslash cdots+u\text{'}\_ny^\{(n-2)\}\_n$ the parameters commute out, with a little "dirt": :$f=u\_1P(D)y\_1+u\_2P(D)y\_2+\backslash cdots+u\_nP(D)y\_n+u\text{'}\_1y^\{(n-1)\}\_1+u\text{'}\_2y^\{(n-1)\}\_2+\backslash cdots+u\text{'}\_ny^\{(n-1)\}\_n.$ But $P(D)y\_j=0$, therefore :$f=u\text{'}\_1y^\{(n-1)\}\_1+u\text{'}\_2y^\{(n-1)\}\_2+\backslash cdots+u\text{'}\_ny^\{(n-1)\}\_n.$ This, with the constraints, gives a linear system in the $u\text{'}\_j$. This much can always be solved; in fact, combining Cramer's Rule with the Wronskian , : The rest is a matter of integrating $u\text{'}\_j.$ The particular solution is not unique; $y\_p+c\_1y\_1+\backslash cdots+c\_ny\_n$ also satisfies the ODE for any set of constants ''cj''. See also Variation Of Parameters . Example: Suppose $y\text{'}\text{'}-4y\text{'}+5y=sin(kx)$. We take the solution basis found above $\backslash \{e^\{(2+i)x\},e^\{(2-i)x\}\backslash \}$. Using the List Of Integrals Of Exponential Functions where the Wronskian

  U - \int { an xdx = - \ln \left {\sec x} ight + C} \
  Y P f = uy_1 + vy_2 = \cos x\ln \left {\cos x} ight + x\sin x \
  {{ExampleSidebar35%<math>y'+3y 2 \,</math>