Difference between revisions of "Differential equation"
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==Examples== | ==Examples== | ||
− | <math>f(x) = f'(x)</math> has solutions <math>Ce^x</math> for all real constants <math>C</math>. | + | <math>f(x) = f'(x)</math> has solutions <math>Ce^x</math> for all real constants <math>C</math>. With the initial condition <math>f(0) = 1</math>, <math>f(x) = e^x</math> becomes the unique solution. |
− | <math>f(x) = -f''(x)</math> has solutions <math>C \cos(x + a)</math> for all real constants <math>C</math> and <math>a</math>. The solutions with <math>a = 0</math> are <math>C \cos x</math>; those with <math>a = \frac{\pi}{2}</math> are <math>C \sin x</math>. | + | <math>f(x) = -f''(x)</math> has solutions <math>C \cos(x + a)</math> for all real constants <math>C</math> and <math>a</math>. The solutions with <math>a = 0</math> are <math>C \cos x</math>; those with <math>a = -\frac{\pi}{2}</math> are <math>C \sin x</math>. The initial condition <math>f'(0) = 0</math> produces the cosine solutions; <math>f(0) = 0</math> produces the sine solutions. |
==Solutions== | ==Solutions== |
Revision as of 10:40, 2 May 2022
A differential equation is a functional equation involving functions and their derivatives.
The order of a differential equation is the largest order of any derivative that appears in the equation.
Differential equations are often presented along with an initial condition; that is, a given value of the function or dependent variable at some value of its independent variable.
Examples
has solutions
for all real constants
. With the initial condition
,
becomes the unique solution.
has solutions
for all real constants
and
. The solutions with
are
; those with
are
. The initial condition
produces the cosine solutions;
produces the sine solutions.
Solutions
Separation of variables is a convenient technique for solving certain types of differential equations. Essentially, the method involves rewriting the equation so that each side is an expression in only one variable and then taking the antiderivative of both sides.
When solving differential equations, it is best to notate functions using a single variable name instead of spelling out the function and its arguments, for example using instead of
. Here we also use Leibniz notation
for the derivative because it allows for manipulating
and
individually.
Worked example
To solve the differential equation we move all terms containing
and
to the right and all terms containing
and
to the left, thus obtaining
and
are now factors on opposite sides, so we can antidifferentiate both sides:
The right integral is simply
for some constant
.
Using partial fractions, basic integration rules, and identities of the logarithm and absolute value functions, the left side becomes again for some constant
.
Constants of integration and
can be combined into a single constant
(this generally happens in separation of variables), so we write
When is in the range
, algebraic manipulation leads to the solution
Given any initial condition, we can solve for the value of the constant
.
Approximations
Euler's method uses repeated tangent-line approximations to approximate a value of the solution to a first-order differential equation given an initial condition
.
If , Euler's method works by subdividing
into smaller intervals
, sometimes called steps. Starting at
, for each step
, the value of
(at the end of the step) is approximated via a tangent line about
(the beginning of the step, where
is known and
can be computed in terms of
and
using the given differential equation), until
is reached.
The formula for the tangent-line approximation is
The quantity is called the step size. Euler's method can be employed when
by simply using negative step sizes.
Constant expressions
Certain expressions involving solutions to differential equations can be proven constant by noting that their derivatives are always . These constant expressions can then be used to prove properties of the solutions.
For example, when ,
Using
allows for reconstructing the familiar identity
for all real
.
When , for any real constant
,
Letting
and evaluating at both
and
gives
which using
becomes the familiar identity
for all real
and
.
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