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Khan: What is a differential equation?

Differential Equations

Khan D.E. Lesson 01 :

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A Differential Equation

A differential equation is an equation relating derivatives of a function to the function itself.

The solution to a differential equation is a function.

Three ways to express the same differential equation:

$\displaystyle{y^{\prime\prime} + 2y^\prime = 3y}$

$\displaystyle{\frac{d^2y}{dx^2} + 2\frac{dy}{dx} = 3y}$

$\displaystyle{f^{\prime\prime}(x) + 2f^\prime(x) = 3f(x)}$

There may be more than one solution to the same differential equation.

In fact, the usual thing is that there are several solutions including classes of functions which vary only by a constant term.

The equation above has (at least) two solutions. One suggested solution is $y = e^{-3x}$.

$\displaystyle{ y = e^{-3x} }$

$\displaystyle{ \frac{dy}{dx} = -3e^{-3x}}$

$\displaystyle{ \frac{d^2y}{dx^2} = 9e^{-3x}}$

which makes the differential equation hold.

Another solution is $y=e^x$.

$\displaystyle{ y = e^x }$

$\displaystyle{ \frac{dy}{dx} = e^x}$

$\displaystyle{ \frac{d^2y}{dx^2} = e^x}$

which also solves the equation.

Ordinary vs. partial differential equations.

A partial differential equation has at least one partial derivative. but if it has only ordinary differentials, it is an ordinary differential equation. Ordinary differential equations are the stuff of all first courses.

Order of differential equations.

The order of a differential equation is the number of the highest derivative, for example, if it is $ \frac{d^2y}{dx^2}$ the equation is second order or if it is $ \frac{dy}{dx}$ the equation is first order.