Ordinary Differential Equation

An ordinary differential equation (ODE) is a differential equation for a function of one variable. (This differs from the partial differential equation , where the unknown is a function of several variables.) Thus, ODEs are equations of the form

F(x,y,y',y'',...,y^{(n)})=0,\qquad (1)

{\ displaystyle F (x, y, y ', y' ', ..., y ^ {(n)}) = 0, \ qquad (1)}

F (x, y, y ', y' ', ..., y ^ {{(n)}}) = 0, \ qquad (1)

Where $y(x)$ ${\ displaystyle y (x)}$ $y (x)$ Is an unknown function (possibly a vector function , then $F$ ${\ displaystyle F}$ $F$ , as a rule, also a vector function with values in a space of the same dimension ; in this case we speak of a system of differential equations), depending on an independent variable $x$ ${\ displaystyle x}$ $x$ , prime means differentiation by $x$ ${\ displaystyle x}$ $x$ . Number $n$ ${\ displaystyle n}$ $n$ (the order of the highest derivative included in this equation) is called the order of differential equation (1).

Independent variable $x$ ${\ displaystyle x}$ $x$ it is often interpreted (especially in differential equations arising in physical and other natural-scientific problems) as time , therefore it is often denoted by the letter $t$ ${\ displaystyle t}$ $t$ . Variable $y$ ${\ displaystyle y}$ $y$ - some quantity (or a set of values if $y$ ${\ displaystyle y}$ $y$ is a vector function) that changes over time. For example, $y$ ${\ displaystyle y}$ $y$ may mean a set of coordinates of a point in space; in this case, equation (1) describes the motion of a point in space, that is, the change in its coordinates over time. Independent variable $x$ ${\ displaystyle x}$ $x$ usually takes real values, but differential equations in which the variable $x$ ${\ displaystyle x}$ $x$ complex (the so-called equations with complex time ).

The most common differential equations of the form

y^{(n)}=f(x,y,y',y'',...,y^{(n-1)}),\qquad (2)

{\ displaystyle y ^ {(n)} = f (x, y, y ', y' ', ..., y ^ {(n-1)}), \ qquad (2)}

y ^ {{(n)}} = f (x, y, y ', y' ', ..., y ^ {{(n-1)}}), \ qquad (2)

in which the highest derivative $y^{(n)}$ ${\ displaystyle y ^ {(n)}}$ $y ^ {{(n)}}$ expressed as a function of variables $x,$ ${\ displaystyle x,}$ $x$ $y$ ${\ displaystyle y}$ $y$ and derivatives $y^{(i)}$ ${\ displaystyle y ^ {(i)}}$ ${\ displaystyle y ^ {(i)}}$ orders less $n .$ ${\ displaystyle n.}$ $n$ Such differential equations are called normal or resolved with respect to the derivative .

In contrast to equations of the form (2), differential equations of the form (1) are called equations that are not resolved with respect to the derivative or implicit differential equations.

The classical solution of differential equation (2) is called $n$ ${\ displaystyle n}$ $n$ differentiable function $y(x)$ ${\ displaystyle y (x)}$ $y (x)$ satisfying the equation at all points of its domain of definition . Usually, there are many such functions, and to select one of them you need to impose an additional condition on it. The initial condition for equation (2) is the condition

y(x_{0})=y_{0},\ y'(x_{0})=y_{0}^{(1)},y''(x_{0})=y_{0}^{(2)},\,\ldots ,\,y^{(n-1)}(x_{0})=y_{0}^{(n-1)},\qquad (3)

{\ displaystyle y (x_ {0}) = y_ {0}, \ y '(x_ {0}) = y_ {0} ^ {(1)}, y' '(x_ {0}) = y_ {0 } ^ {(2)}, \, \ ldots, \, y ^ {(n-1)} (x_ {0}) = y_ {0} ^ {(n-1)}, \ qquad (3)}

y (x_ {0}) = y_ {0}, \ y '(x_ {0}) = y_ {0} ^ {{(1)}}, y' '(x_ {0}) = y_ {0} ^ {{((2)}}, \, \ ldots, \, y ^ {{(n-1)}} (x_ {0}) = y_ {0} ^ {{(n-1)}}, \ qquad (3)

Where $x_{0}$ ${\ displaystyle x_ {0}}$ $x_ {0}$ - some fixed value of an independent variable (fixed point in time), and $y_{0}$ ${\ displaystyle y_ {0}}$ $y_0$ and $y_{0}^{(i)}$ ${\ displaystyle y_ {0} ^ {(i)}}$ $y_ {0} ^ {{(i)}}$ - accordingly, the fixed values of the function $y$ ${\ displaystyle y}$ $y$ and all its derivatives to order $n-1$ ${\ displaystyle n-1}$ $n-1$ inclusive. Differential equation (2) together with the initial condition (3) is called the initial problem or the Cauchy problem :

\left\{{\begin{array}{lcl}y^{(n)}=f(x,y,y',y'',...,y^{(n-1)}),\\{}\\y(x_{0})=y_{0},\ y'(x_{0})=y_{0}^{(1)},y''(x_{0})=y_{0}^{(2)},\,\ldots ,\,y^{(n-1)}(x_{0})=y_{0}^{(n-1)}.\end{array}}\right.

{\ displaystyle \ left \ {{\ begin {array} {lcl} y ^ {(n)} = f (x, y, y ', y' ', ..., y ^ {(n-1)} ), \\ {} \\ y (x_ {0}) = y_ {0}, \ y '(x_ {0}) = y_ {0} ^ {(1)}, y' '(x_ {0} ) = y_ {0} ^ {(2)}, \, \ ldots, \, y ^ {(n-1)} (x_ {0}) = y_ {0} ^ {(n-1)}. \ end {array}} \ right.}

\ left \ {{\ begin {array} {lcl} y ^ {{(n)}} = f (x, y, y ', y' ', ..., y ^ {{(n-1)} }), \\ {} \\ y (x_ {0}) = y_ {0}, \ y '(x_ {0}) = y_ {0} ^ {{(1)}}, y' '(x_ {0}) = y_ {0} ^ {{(2)}}, \, \ ldots, \, y ^ {{(n-1)}} (x_ {0}) = y_ {0} ^ {{ (n-1)}}. \ end {array}} \ right.

Picard's theorem states that under fairly general restrictions on the function $f$ ${\ displaystyle f}$ $f$ standing on the right side of equation (2), the Cauchy problem for this equation has a unique solution defined on a certain interval of the time axis $x$ ${\ displaystyle x}$ $x$ containing the initial value $x_{0}$ ${\ displaystyle x_ {0}}$ $x_ {0}$ (this interval, generally speaking, may not coincide with the entire axis).

The main tasks and results of the theory of differential equations: the existence and uniqueness of solving various problems for ODEs, methods for solving the simplest ODEs , a qualitative study of the solutions of ODEs without finding their explicit form.

History

Differential equations were already found in the works of I. Newton and G. Leibniz ; the term "differential equations" belongs to Leibniz. When creating the calculus of "fluxia" and "fluent", Newton posed two tasks: from this ratio between fluents, determine the ratio between fluxes; using this equation containing fluxia, find the ratio between the fluents. From the modern point of view, the first of these problems (calculation by functions of their derivatives) relates to differential calculus, and the second is the content of the theory of ordinary differential equations. Newton considered the problem of finding the indefinite integral F (x) of the function f (x) simply as a special case of his second problem. This approach was justified for Newton as the creator of the foundations of mathematical science: in a very large number of cases, the laws of nature that govern certain processes are expressed in the form of differential equations, and the calculation of the course of these processes is reduced to solving the differential equation. ^[one]

The main discovery of Newton, which he considered it necessary to classify and published only as an anagram, is as follows: "Data aequatione quotunque fluentes quantitae involvente fluxiones invenire et vice versa." Translated into modern mathematical language, this means: "It is useful to solve differential equations." At present, the theory of differential equations is a difficultly visible conglomerate of a large number of diverse ideas and methods, highly useful for all kinds of applications and constantly stimulating theoretical research in all departments of mathematics. ^[2] ^[3]

Examples

One of the simplest applications of differential equations is to solve the non-trivial problem of finding the trajectory of a body from known acceleration projections. For example, in accordance with Newton’s second law, the acceleration of a body is proportional to the sum of the acting forces; the corresponding differential equation has the form $m{\ddot {x}}=F(x,t)$ ${\ displaystyle m {\ ddot {x}} = F (x, t)}$ . Knowing the acting forces (the right-hand side), we can solve this equation and, taking into account the initial conditions (coordinates and speed at the initial moment of time), find the trajectory of the point.
Differential equation $y'=y$ ${\ displaystyle y '= y}$ , together with the initial condition $y(0)=1$ ${\ displaystyle y (0) = 1}$ , sets the exponent : $y(x)=e^{x}$ ${\ displaystyle y (x) = e ^ {x}}$ . If a $x$ ${\ displaystyle x}$ denotes time, then this function describes, for example, population growth in conditions of unlimited resources, as well as much more.
By solving a differential equation $y'=f(x)$ ${\ displaystyle y '= f (x)}$ , the right side of which does not depend on an unknown function, is an indefinite integral

y(x)=\int \!f(x)\,dx+C,

{\ displaystyle y (x) = \ int \! f (x) \, dx + C,}

Where $C$ ${\ displaystyle C}$ Is an arbitrary constant.

First-Order Differential Equations

Separated Variable Equations

Differential equation ${\dot {y}}=f(x,y)$ ${\ displaystyle {\ dot {y}} = f (x, y)}$ is called an equation with separable variables if its right-hand side is representable in the form $y'=f_{1}(x)f_{2}(y)$ ${\ displaystyle y '= f_ {1} (x) f_ {2} (y)}$ . Then, in the case $f_{2}(y)\neq 0$ ${\ displaystyle f_ {2} (y) \ neq 0}$ , the general solution to the equation is $\int \!{\frac {dy}{f_{2}(y)}}=\int \!f_{1}(x)\,dx$ ${\ displaystyle \ int \! {\ frac {dy} {f_ {2} (y)}} = \ int \! f_ {1} (x) \, dx}$ .

Examples of physical problems leading to equations with separable variables

Body Cooling

Let be $T$ ${\ displaystyle T}$ - Body temperature, $T_{0}$ ${\ displaystyle T_ {0}}$ - ambient temperature ( $T>T_{0}$ ${\ displaystyle T> T_ {0}}$ ) Let be $Q$ ${\ displaystyle Q}$ - the amount of heat $c$ ${\ displaystyle c}$ - specific heat . Then the amount of heat transferred to the environment before the temperature equalizes is expressed by the formula $Q=mc(T-T_{0})$ ${\ displaystyle Q = mc (T-T_ {0})}$ , or, in differential form, $dQ=mc\,dT$ ${\ displaystyle dQ = mc \, dT}$ . On the other hand, the rate of heat transfer can be expressed as $dQ=-k(T-T_{0})\,dt$ ${\ displaystyle dQ = -k (T-T_ {0}) \, dt}$ where $k$ ${\ displaystyle k}$ - a certain coefficient of proportionality. Excluding from these two equations $d Q$ ${\ displaystyle dQ}$ , we obtain an equation with separable variables:

mc\,dT=-k(T-T_{0})\,dt

{\ displaystyle mc \, dT = -k (T-T_ {0}) \, dt}

.

A general solution to this equation is a family of functions $T=T_{0}+Ce^{-{\frac {kt}{mc}}}$ ${\ displaystyle T = T_ {0} + Ce ^ {- {\ frac {kt} {mc}}}}$ .

Homogeneous equations

Differential equation ${\dot {y}}=f(x,y)$ ${\ displaystyle {\ dot {y}} = f (x, y)}$ called homogeneous if $f(x,y)$ ${\ displaystyle f (x, y)}$ Is a homogeneous function of degree zero. Function $f(x,y)$ ${\ displaystyle f (x, y)}$ called homogeneous degree $k$ ${\ displaystyle k}$ if for any $\lambda >0$ ${\ displaystyle \ lambda> 0}$ equality holds $f(\lambda x,\lambda y)=\lambda ^{k}f(x,y)$ ${\ displaystyle f (\ lambda x, \ lambda y) = \ lambda ^ {k} f (x, y)}$ .

Replacement $y(x)=xz(x)$ ${\ displaystyle y (x) = xz (x)}$ leads when $x>0$ ${\ displaystyle x> 0}$ homogeneous equation to the equation with separable variables:

f(x,xz)=x^{0}f(1,z)=f(1,z)

{\ displaystyle f (x, xz) = x ^ {0} f (1, z) = f (1, z)}

{\dot {y}}=x{\dot {z}}+z

{\ displaystyle {\ dot {y}} = x {\ dot {z}} + z}

Substituting in the original equation, we obtain:

{\dot {z}}={\frac {1}{x}}(f(1,z)-z)

{\ displaystyle {\ dot {z}} = {\ frac {1} {x}} (f (1, z) -z)}

,

which is an equation with separable variables.

Quasihomogeneous equations

Differential equation ${\dot {y}}=f(x,y)$ ${\ displaystyle {\ dot {y}} = f (x, y)}$ called quasihomogeneous if for any $\lambda >0$ ${\ displaystyle \ lambda> 0}$ the relation holds $f\left(\lambda ^{\alpha }x,\lambda ^{\beta }y\right)=\lambda ^{\beta -\alpha }f(x,y)$ ${\ displaystyle f \ left (\ lambda ^ {\ alpha} x, \ lambda ^ {\ beta} y \ right) = \ lambda ^ {\ beta - \ alpha} f (x, y)}$ .

This equation is solved by replacing $y=z^{\frac {\beta }{\alpha }}$ ${\ displaystyle y = z ^ {\ frac {\ beta} {\ alpha}}}$ :

{\dot {z}}={\frac {\alpha }{\beta }}\left(z^{-{\frac {1}{\alpha }}}\right)^{\beta -\alpha }f\left(x,z^{\frac {\beta }{\alpha }}\right)

{\ displaystyle {\ dot {z}} = {\ frac {\ alpha} {\ beta}} \ left (z ^ {- {\ frac {1} {\ alpha}}} \ right) ^ {\ beta - \ alpha} f \ left (x, z ^ {\ frac {\ beta} {\ alpha}} \ right)}

Due to quasihomogeneity, putting $\lambda =z^{-{\frac {1}{\alpha }}}$ ${\ displaystyle \ lambda = z ^ {- {\ frac {1} {\ alpha}}}}$ we get:

\left(z^{-{\frac {1}{\alpha }}}\right)^{\beta -\alpha }f\left(x,z^{\frac {\beta }{\alpha }}\right)=f\left({\frac {x}{z}},1\right)

{\ displaystyle \ left (z ^ {- {\ frac {1} {\ alpha}}} \ right) ^ {\ beta - \ alpha} f \ left (x, z ^ {\ frac {\ beta} {\ alpha}} \ right) = f \ left ({\ frac {x} {z}}, 1 \ right)}

{\dot {z}}={\frac {\alpha }{\beta }}f\left({\frac {x}{z}},1\right)

{\ displaystyle {\ dot {z}} = {\ frac {\ alpha} {\ beta}} f \ left ({\ frac {x} {z}}, 1 \ right)}

,

which is obviously a homogeneous equation.

Linear Equations

Differential equation $y'+a(x)y=b(x)$ ${\ displaystyle y '+ a (x) y = b (x)}$ called linear and can be solved by three methods: the method of integrating factor, the method of variation of the constant, or the Bernoulli method.

Integrating Multiplier Method

Let a function be given $\mu (x)$ ${\ displaystyle \ mu (x)}$ - integrating factor, in the form:

\mu (x)=e^{\int \!a(x)\,dx}

{\ displaystyle \ mu (x) = e ^ {\ int \! a (x) \, dx}}

Multiply both sides of the original equation by $\mu (x)$ ${\ displaystyle \ mu (x)}$ we get:

{\dot {y}}e^{\int \!a(x)\,dx}+ya(x)e^{\int \!a(x)\,dx}=b(x)\mu (x)

{\ displaystyle {\ dot {y}} e ^ {\ int \! a (x) \, dx} + ya (x) e ^ {\ int \! a (x) \, dx} = b (x) \ mu (x)}

It is easy to see that the left side is a derivative of the function $\mu (x)y(x)$ ${\ displaystyle \ mu (x) y (x)}$ by $x$ ${\ displaystyle x}$ . Therefore, the equation can be rewritten:

(\mu (x)y(x))'=b(x)\mu (x)

{\ displaystyle (\ mu (x) y (x)) '= b (x) \ mu (x)}

Integrate:

y(x)\mu (x)=\int \!b(x)\mu (x)\,dx+C

{\ displaystyle y (x) \ mu (x) = \ int \! b (x) \ mu (x) \, dx + C}

Thus, the solution of the linear equation will be:

y(x)=e^{-\int \!a(x)\,dx}\left(\int \!b(x)\mu (x)\,dx+C\right)

{\ displaystyle y (x) = e ^ {- \ int \! a (x) \, dx} \ left (\ int \! b (x) \ mu (x) \, dx + C \ right)}

Constant Variation Method (Lagrange Method)

We consider the homogeneous equation ${\dot {y}}+a(x)y=0$ ${\ displaystyle {\ dot {y}} + a (x) y = 0}$ . Obviously, this is an equation with separable variables, its solution:

y(x)=ce^{-\int \!a(x)\,dx}

{\ displaystyle y (x) = ce ^ {- \ int \! a (x) \, dx}}

We will search for solutions of the original equation in the form:

y(x)=c(x)e^{-\int \!a(x)\,dx}

{\ displaystyle y (x) = c (x) e ^ {- \ int \! a (x) \, dx}}

Substituting the resulting solution in the original equation:

{\dot {c}}=b(x)e^{\int \!a(x)\,dx}

{\ displaystyle {\ dot {c}} = b (x) e ^ {\ int \! a (x) \, dx}}

,

we get:

c(x)=c_{1}+\int \!b(x)e^{\int \!a(x)\,dx}\,dx

{\ displaystyle c (x) = c_ {1} + \ int \! b (x) e ^ {\ int \! a (x) \, dx} \, dx}

,

Where $c_{1}$ ${\ displaystyle c_ {1}}$ Is an arbitrary constant.

Thus, the solution to the original equation can be obtained by substituting $c(x)$ ${\ displaystyle c (x)}$ into the solution of the homogeneous equation:

y(x)=e^{-\int \!a(x)\,dx}\left(c_{1}+\int \!b(x)e^{\int \!a(x)\,dx}\,dx\right)

{\ displaystyle y (x) = e ^ {- \ int \! a (x) \, dx} \ left (c_ {1} + \ int \! b (x) e ^ {\ int \! a (x ) \, dx} \, dx \ right)}

Bernoulli equation

Differential equation ${\dot {y}}+a(x)y=b(x)y^{n}$ ${\ displaystyle {\ dot {y}} + a (x) y = b (x) y ^ {n}}$ called the Bernoulli equation (for $n=0$ ${\ displaystyle n = 0}$ or $n=1$ ${\ displaystyle n = 1}$ we obtain an inhomogeneous or homogeneous linear equation). At $n=2$ ${\ displaystyle n = 2}$ is a special case of the Riccati equation . Named after Jacob Bernoulli , who published this equation in 1695 . A solution method using a replacement that reduces this equation to a linear one was found by his brother Johann Bernoulli in 1697 .

Binomial differential equation

This is an equation of the form

\left(y'\right)^{m}=f(x,y),

{\ displaystyle \ left (y '\ right) ^ {m} = f (x, y),}

Where

m

{\ displaystyle m}

Is a natural number , and

f(x,y)

{\ displaystyle f (x, y)}

- polynomial in two variables ^[4] .

Literature

Tutorials

Arnold V. I. Ordinary Differential Equations, - Any Edition.
Arnold V. I. Additional chapters of the theory of ordinary differential equations, - Any

edition.

Arnold V.I. Geometric methods in the theory of ordinary differential equations, - Any publication.
Arnold V.I. , Ilyashenko Yu. S. Ordinary differential equations, - Itogi Nauki i Tekhniki. Ser. Modern prob. mat. Fundam. Direction., 1985, Volume 1.
Petrovsky I. G. Lectures on the theory of ordinary differential equations, - Any publication.
Pontryagin L. S. Ordinary differential equations, - Any publication.
Stepanov VV Course of differential equations, - Any publication.
Tricomi F. Differential Equations, - Any Edition.
Filippov AF Introduction to the theory of differential equations, - Any publication.
Philips G. Differential Equations, - Any Edition.
Hartman F. Ordinary Differential Equations, - Any Edition.
Elsgolts L. E. Differential equations and calculus of variations - Any publication.

Bookkeepers

Filippov A.F. Collection of problems on differential equations, - Any publication.

Directories

Kamke E. Handbook of ordinary differential equations, - Any publication.
Zaitsev V.F., Polyanin A.D. Handbook of ordinary differential equations, - Any publication.

Notes

↑ TSB. Differential equations
↑ Arnold V.I. Additional chapters of the theory of ordinary differential equations.
↑ Arnold V.I. Geometric methods in the theory of ordinary differential equations.
↑ Zwillinger, D. Handbook of Differential Equations. - 3rd ed. - Academic Press, 1997 .-- P. 120.

[1] TSB. Differential equations

[2] Arnold V.I. Additional chapters of the theory of ordinary differential equations.

[3] Arnold V.I. Geometric methods in the theory of ordinary differential equations.

[4] Zwillinger, D. Handbook of Differential Equations. - 3rd ed. - Academic Press, 1997 .-- P. 120.