Differential equations
Differential equations are functional equations 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
Approximations
Euler's method uses repeated tangent-line approximations to approximate the solution to first-order differential equations.
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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