Differential equation - Revision history

Vipul: /* Terminology */

2024-04-18T06:56:43Z

Terminology

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 | solution to initial value problem || a particular solution that satisfies the initial value condition. || A particular solution to <math>y + y' + y'' = (x + 1)^2</math> satisfying <math>y(0) = -1, y'(0) = 0</math> (i.e., <math>y = -1</math> and <math>y' = 0</math> when <math>x = 0</math>) is <math>y = x^2 - 1</math>.
 |}

Vipul: /* Key observations */

2024-04-18T06:11:55Z

Key observations

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	Functional equations involving derivatives that do ''not'' fit this definition of differential equation are also studied, but the study of these is more complicated and requires new techniques. [[Delay differential equation]]s is one such class of functional equations.		Functional equations involving derivatives that do ''not'' fit this definition of differential equation are also studied, but the study of these is more complicated and requires new techniques. [[Delay differential equation]]s is one such class of functional equations.

			===Functional equations involving iterated derivatives at a single point can be simplied to differential equations===

			Consider an equation like this:

			<math>(yy''\sin(ye^x + xy'))' = yx^3 + x\cos y</math>

			Here, all derivatives are with respect to <math>x</math>.

			This does not meet the standard format of a differential equation, because it involves taking a derivative of an intermediate expression <math>yy''\sin(ye^x + xy')</math>, whereas a standard format differential equation only allows for derivatives (including repeated derivatives) of the dependent variable <math>y</math>.

			However, this can be ''simplified'' to a differential equation in the standard format, specifically by using various differentiation rules such as the product rule, chain rule, and the linearity of differentiation, plus the formulas for differentiating the sine and exponential functions. Therefore, we generally consider this sort of equation to be an ordinary differential equation, albeit in an unsimplified form.

			More abstractly, if <math>G</math> is a function of <math>x, y, y', \dots, y^{(k)}</math>, then <math>G'</math> can be written as follows using [[partial derivative]]s by using the [[chain rule for partial differentiation]]:

			<math>G' = G_x + G_y y' + G_{y'} y'' + \dots G_{y^{(k)}} y^{(k+1)}</math>

			In particular, this allows us to simplify the intermediate derivative expressions into expressions in <math>x, y, y', \dots, y^{(k + 1)}</math> and thereby allows us to convert the differential equation to a standard format.

			(Note that in any particular case such as the example above, we don't need to think in terms of partial derivatives, but the abstract formulation does require the use of partial derivatives).

	===It does not make sense to ask whether a point satisfies a differential equation===		===It does not make sense to ask whether a point satisfies a differential equation===

Vipul: /* Relation with system of first-order differential equations */

2024-04-11T02:30:18Z

Relation with system of first-order differential equations

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	==Relation with system of first-order differential equations==		==Relation with system of first-order differential equations==

	Any differential equation of order <math>k</math> can be converted to a [[system of first-order differential equations]] with <math>k</math> equations and <math>k</math> variables (i.e., <math>k</math> unknown functions that we are trying to solve for). For the conversion procedure, see [[conversion of a differential equation to a system of first-order differential equations]]. However, the converse is not true, i.e., it is not always possible to convert a system of first-order differential equations with multiple dependent ~~variable~~ into a single differential equation of higher order with one dependent variable.		Any differential equation of order <math>k</math> can be converted to a [[system of first-order differential equations]] with <math>k</math> equations and <math>k</math> variables (i.e., <math>k</math> unknown functions that we are trying to solve for). For the conversion procedure, see [[conversion of a differential equation to a system of first-order differential equations]]. However, the converse is not true, i.e., it is not always possible to convert a system of first-order differential equations with multiple dependent variables into a single differential equation of higher order with one dependent variable.

	The same idea can be used to perform the [[conversion of a system of differential equations to a system of first-order differential equations]].		The same idea can be used to perform the [[conversion of a system of differential equations to a system of first-order differential equations]].

	==Solution strategies==		==Solution strategies==

Vipul: /* Solution terminology */

2021-09-24T23:23:18Z

Solution terminology

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	===Solution terminology===		===Solution terminology===

	~~<center>{{#widget:YouTube\|id=YbsgAjNKorw}}</center>~~

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Vipul: /* Equation terminology */

2021-09-24T23:22:48Z

Equation terminology

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	===Equation terminology===		===Equation terminology===

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Vipul: /* Key observations */

2021-09-24T23:22:35Z

Key observations

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	==Key observations==		==Key observations==

	~~<center>{{#widget:YouTube\|id=NMCF-ORnkho}}</center>~~

	===Differential equations are functional equations===		===Differential equations are functional equations===

Vipul: /* Definition */

2021-09-24T23:22:27Z

Definition

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	==Definition==		==Definition==

	~~<center>{{#widget:YouTube\|id=lv_pOVWfg0o}}</center>~~

	===Formal description===		===Formal description===

Vipul at 21:15, 25 July 2012

2012-07-25T21:15:55Z

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	==Definition==		==Definition==

			<center>{{#widget:YouTube\|id=lv_pOVWfg0o}}</center>

	===Formal description===		===Formal description===
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	==Key observations==		==Key observations==

			<center>{{#widget:YouTube\|id=NMCF-ORnkho}}</center>

	===Differential equations are functional equations===		===Differential equations are functional equations===
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	===Equation terminology===		===Equation terminology===

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	===Solution terminology===		===Solution terminology===

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Vipul: /* Solution strategies */

2012-07-09T21:03:43Z

Solution strategies

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	\| [[homogeneous linear differential equation with constant coefficients]] \|\| any \|\| 1 \|\| We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.		\| [[homogeneous linear differential equation with constant coefficients]] \|\| any \|\| 1 \|\| We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.
	\|}		\|}

			===Qualitative methods===

			For most differential equations, it is very hard to convert the differential equation to a series of integration problems and to find explicit expressions for the solution. Instead, in many cases, we try to determine the qualitative properties of solution functions, including existence, uniqueness, extent of differentiability, nature of roots and critical points, etc.

Vipul: /* Solution strategies in particular cases */

2012-07-09T21:01:46Z

Solution strategies in particular cases

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	\| [[first-order linear differential equation]] which in simplified form looks like <math>y' + p(x)y = q(x)</math> \|\| 1 \|\| 1 \|\| Use the [[integrating factor]] <math>e^{H(x)}</math> where <math>H'=p</math>. The general solution is <math>y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx</math>		\| [[first-order linear differential equation]] which in simplified form looks like <math>y' + p(x)y = q(x)</math> \|\| 1 \|\| 1 \|\| Use the [[integrating factor]] <math>e^{H(x)}</math> where <math>H'=p</math>. The general solution is <math>y = Ce^{-H(x)} + e^{-H(x)}\int p(x)e^{H(x)} \, dx</math>
	\|-		\|-
	\| [[separable differential equation]] which is of the form <math>y' = f(x)g(y)</math> (any first-order [[autonomous differential equation]] is separable, though there are separable differential equations that aren't autonomous) \|\| 1 \|\| 1 \|\| Separate and solve as <math>\int \frac{dy}{g(y)} = \int f(x) \, dx</math>. Also find solutions corresponding to <math>y = k</math> where <math>g(k) = 0</math>.		\| [[separable differential equation]] which is of the form <math>y' = f(x)g(y)</math> (any [[first-order first-degree autonomous differential equation]] is separable, though there are separable differential equations that aren't autonomous) \|\| 1 \|\| 1 \|\| Separate and solve as <math>\int \frac{dy}{g(y)} = \int f(x) \, dx</math>. Also find solutions corresponding to <math>y = k</math> where <math>g(k) = 0</math>.
			\|-
			\| [[first-order exact differential equation]] <math>F(x,y,y') = 0</math> \|\| 1 \|\| 1 \|\| Try to find a relation <math>R(x,y)</math> such that <math>F(x,y,y') = \frac{d}{dx}[R(x,y)]</math> using [[implicit differentiation]]. Finding the <math>R</math>, even if it does exist, can be tricky.
			\|-
			\| [[Bernoulli differential equation]] <math>y' + p(x)y = q(x)y^n</math> (<math>n \ne 0,1</math>) \|\| 1 \|\| 1 \|\| Divide both sides by <math>y^n</math> (set aside possible stationary solution <math>y = 0</math>), then substitute <math>w = 1/y^{n-1}</math> to get a [[first-order linear differential equation]] with dependent variable <math>w</math> and independent variable <math>x</math>.
	\|-		\|-
	\| [[Clairaut's equation]] which is of the form <math>y = xy' + f(y')</math> \|\| 1 \|\| need not be polynomial; if polynomial, may have any degree \|\| <math>y = Cx + f(C)</math> with <math>C \in \R</math> (all straight lines) and a single other solution explicitly described as the solution to <math>x + f(dy/dx) = 0</math>, given by <math>x = -f'(p), y = f(p) - pf'(p)</math> as a parametric curve in terms of <math>p</math>.		\| [[Clairaut's equation]] which is of the form <math>y = xy' + f(y')</math> \|\| 1 \|\| need not be polynomial; if polynomial, may have any degree \|\| <math>y = Cx + f(C)</math> with <math>C \in \R</math> (all straight lines) and a single other solution explicitly described as the solution to <math>x + f(dy/dx) = 0</math>, given by <math>x = -f'(p), y = f(p) - pf'(p)</math> as a parametric curve in terms of <math>p</math>.
			\|-
			\| [[Lagrange equation]] <math>y = f(y')x + g(y')</math> which is linear in <math>x</math> and <math>y</math> but not necessarily in <math>y'</math> \|\| 1 \|\| need not be polynomial; if polynomial, may have any degree \|\| General solution is a family of curves, each described as a parametric curve with parameter the derivative <math>y'</math> (which we denote by <math>p</math>). There may be some special straight line solutions of the form <math>y = px + g(p)</math> for values <math>p</math> with <math>p = f(p)</math>.
	\|-		\|-
	\| [[second-order autonomous differential equation of degree one]], which is of the form <math>y'' = F(y,y')</math> \|\| 2 \|\| 1 \|\|		\| [[second-order autonomous differential equation of degree one]], which is of the form <math>y'' = F(y,y')</math> \|\| 2 \|\| 1 \|\|
			\|-
			\| [[homogeneous linear differential equation with constant coefficients]] \|\| any \|\| 1 \|\| We construct the characteristic polynomial of the differential equation, find its real and complex roots, and the space of solution functions is a vector space with basis functions described using these roots.
	\|}		\|}