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.
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.
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.
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 .
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.
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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