Difference between revisions of "Differential equation"
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For example, when <math>f(x) = -f''(x)</math>, <cmath>\begin{align*} \left( f(x)^2 + f'(x)^2 \right)' &= \left( f(x)^2 \right) ' + \left( f'(x)^2 \right) ' \\ &= 2f(x)f'(x) + 2f'(x)f''(x) \\ &= 2f'(x) \left( f(x) + f''(x) \right) \\ &= 0. \end{align*}</cmath> Using <math>\sin' x = \cos x</math> allows for reconstructing the familiar [[Trigonometric identities#Pythagorean identities|identity]] <cmath>\sin^2 x + \cos^2 x = \sin^2 0 + \cos^2 0 = 0^2 + 1^2 = 1</cmath> for all real <math>x</math>. | For example, when <math>f(x) = -f''(x)</math>, <cmath>\begin{align*} \left( f(x)^2 + f'(x)^2 \right)' &= \left( f(x)^2 \right) ' + \left( f'(x)^2 \right) ' \\ &= 2f(x)f'(x) + 2f'(x)f''(x) \\ &= 2f'(x) \left( f(x) + f''(x) \right) \\ &= 0. \end{align*}</cmath> Using <math>\sin' x = \cos x</math> allows for reconstructing the familiar [[Trigonometric identities#Pythagorean identities|identity]] <cmath>\sin^2 x + \cos^2 x = \sin^2 0 + \cos^2 0 = 0^2 + 1^2 = 1</cmath> for all real <math>x</math>. | ||
− | When <math>f(x) = f'(x)</math>, for any real constant <math>S</math>, <cmath>\begin{align*} \left( f(x)f(S-x) \right)' &= f'(x)f(S-x) + f(x)(f(S-x))' \\ &= f'(x)f(S-x) + f(x)(S-x)'f'(S-x) \\ &= f(x)f(S-x) + f(x)(-1)f(S-x) \\ &= 0. \end{align*}</cmath> Letting <math>S = a + b</math> and evaluating at both <math>x = 0</math> and <math>x = a</math> gives <cmath>f( | + | When <math>f(x) = f'(x)</math>, for any real constant <math>S</math>, <cmath>\begin{align*} \left( f(x)f(S-x) \right)' &= f'(x)f(S-x) + f(x)(f(S-x))' \\ &= f'(x)f(S-x) + f(x)(S-x)'f'(S-x) \\ &= f(x)f(S-x) + f(x)(-1)f(S-x) \\ &= 0. \end{align*}</cmath> Letting <math>S = a + b</math> and evaluating at both <math>x = 0</math> and <math>x = a</math> gives <cmath>f(0)f(a+b) = f(a)f(b),</cmath> which using <math>e^0 = 1</math> becomes the familiar [[Exponentiation#Basic Properties|identity]] <cmath>e^{a+b} = e^ae^b</cmath> for all real <math>a</math> and <math>b</math>. |
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[[Category: Calculus]] | [[Category: Calculus]] |
Revision as of 10:03, 5 April 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.
Examples
has solutions
for all real constants
.
has solutions
for all real constants
and
. The solutions with
are
; those with
are
.
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 manipulate and factor to get
then rearrange:
We then proceed to take the antiderivatives
at which point the equation can be solved using integration techniques.
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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