Differential equation

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.

Types of Differential Equations

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

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

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for a simple function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

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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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dy}{G(y)} = F(x)dx }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{dy}{G(y)} = \int F(x)dx }

The solution is then given implicitly by the expression:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 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

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Note the derivative of

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is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(x)e^{\int F(x)} }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\int F(x)} }

The equation can be rewritten

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d(e^{\int F(x)}y)}{dx} = e^{\int F(x)}G(x) }

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\frac{d(e^{\int F(x)}y)}{dx} = \int e^{\int F(x)}G(x) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{\int F(x)}y = \int e^{\int F(x)}G(x) + C }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = e^{-\int F(x)} \int e^{\int F(x)}G(x) + Ce^{-\int F(x)} }

where C is an arbitrary constant.

See also

• Laplace transform
• Separation of variables