Differential equation

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In mathematics, a differential equation is an equation that relates a function to one or more of its derivatives. Differential equations are especially applicable when the tools of algebra, which are ideally suited for static systems, are not enough. Many physical systems are modeled by solving differential equations, although their usefulness extends well into other fields of science such as chemistry and economics.

Contents

Types of Differential Equations

There are two main types of differential equations: Ordinary Differential Equations and Partial Differential Equations.[1]

The former is simpler of the two, as it can be written in the normal form

\frac{d^{(n)}y}{dx^{(n)}} = F(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \frac{d^3y}{dx^3}, ... , \frac{d^{(n-1)}y}{dx^{(n-1)}})

for a simple function y = g(x)

The function F consists of the function y and its derivatives up to the nth order. Notice that y is comprised of only one independent variable x. A differential equation is considered ordinary if the function y in F is dependent on only one variable. It is important to note that, while most ordinary differential equations can be written in the normal form (isolating the highest derivative on one side of the equation and moving all other variables to the other), there are equations in which this cannot be done.


If y were a function of multiple variables, for example

y = F(u,v)

then the derivatives of y in the ordinary equation may be partial derivatives with respect to either u or v. In that case, any differential equation that has partial derivatives is called a partial differential equation. For example, the 1-dimensional wave equation[2] :

\frac{\partial^2 y}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 y}{\partial t^2}

Obviously, partial differential equations are much more complicated to solve.

Methods

There are many ways to find solutions to differential equations.

Ordinary Differential Equations

The simplest differential equations to solve are separable differential equations. A differential equation is separable if it can be written in the form

 \frac{d^{(n)}y}{dx^{(n)}} = F(x)G(y)

Then we can separate the two variables, collect the x's on one side and the y's on the other side, then integrate to get the (n-1) derivative, and integrating again to get the (n-2) derivative, until we have found the function y. For example, for the derivative n = 1:

 \frac{dy}{dx} = F(x)G(y)

\frac{dy}{G(y)} = F(x)dx

\int\frac{dy}{G(y)} = \int F(x)dx

The solution is then given implicitly by the expression:

\int\frac{dy}{G(y)} - \int F(x)dx = C

where C is an arbitrary constant.

Linear Differential Equation Solutions

If a differential equation can be written in the form

F_n(x)\frac{d^{(n)}y}{dx^{(n)}} + F_{n-1}(x)\frac{d^{(n-1)}y}{dx^{(n-1)}} + F_{n-2}(x)\frac{d^{(n-2)}y}{dx^{(n-2)}} + ... + F_1(x)\frac{dy}{dx} + F_0(x)y = G(x)

it is considered a linear differential equation.

First Order Linear Equations

The order of a differential equation is equal to the degree of the highest derivative in the equation. For example, the above equations are order n equations. A first order linear equation appears in the form:

 \frac{dy}{dx} + F(x)y = G(x)

To solve this type of differential equation, an integrating factor[3] is needed. For the first order equation, the integrating factor is defined as

 p = e^{\int F(x)}

Multiplying both sides of the first order equation yields

 \frac{dy}{dx}e^{\int F(x)} + F(x)e^{\int F(x)}y = e^{\int F(x)}G(x)

Note the derivative of

 e^{\int F(x)}

is

 F(x)e^{\int F(x)}

Hence the left hand side of the first order equation now looks like the product rule expansion for

y and  e^{\int F(x)}

The equation can be rewritten

 \frac{d(e^{\int F(x)}y)}{dx} = e^{\int F(x)}G(x)

Now we can integrate both sides, yielding the solution y:

 \int\frac{d(e^{\int F(x)}y)}{dx} = \int e^{\int F(x)}G(x)

 e^{\int F(x)}y = \int e^{\int F(x)}G(x) + C

 y = e^{-\int F(x)} \int e^{\int F(x)}G(x) + Ce^{-\int F(x)}

where C is an arbitrary constant.

References

  1. Edwards, Henry C. and Penney, David E.. Differential Equations and Boundary Value Problems 4th Edition. Upper Saddle River, NJ: Pearson, 2008
  2. Pain, H.J. The Physics of Vibrations and Waves 6th edition. Southern Gate, Chichester, West Sussex, England: John Wiley & Sons, 2005
  3. Edwards, Henry C. and Penney, David E.. Differential Equations and Boundary Value Problems 4th Edition. Upper Saddle River, NJ: Pearson, 2008

See also

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