The independent variable is the variable that the dependent variable is a function of. The derivatives in a DE are taken with respect to the independent variable.
A term is any part of a DE separated by addition or subtraction. Terms are identified by the dependent variable it contains. If a term does not contain the dependent variable, it is called a free term.
A term that is not linear, such as one where the dependent variable or its derivatives are multiplied together, raised to a power other than one, or placed inside a nonlinear function.
A function satisfies a DE when substituting the function and any needed derivatives into the equation makes both sides simplify to the same expression. For example, \(y=2x^2\) satisfies \(xy'-2x^2=y\) since
A general solution of a DE is the general form that all solutions share up to one or more arbitrary constants. For example, \(y=ce^{2x}\) is the general solution for \(y'-2y=0\text{.}\)
A particular solution is a specific function obtained from a general solution by finding or selecting values for its constants. For example, if \(y=ce^{2x}\) is a general solution, then the following are particular solutions:
A family of solutions is the complete collection of all particualr solutions described by a general solution. For example, the family generated by \(y=ce^{2x}\) contains \(y=e^{2x}\text{,}\)\(y=3e^{2x}\text{,}\) and infinitely many other particular solutions corresponding to different values of \(c\text{.}\)
An initial condition is a known value of a solution or one of its derivatives at a specific input value, used to determine the constants in a general solution. For example, \(y(0)=2\) and \(h'(0)=0\) are initial conditions.
A solution curve is the graph of a particular solution. In a family such as \(y=ce^{x^2}+3\text{,}\) each value of \(c\) gives a different curve, and the curve through a chosen initial point is the particular solution for that initial condition.
A constant added when finding an antiderivative or solving a DE by integration. It accounts for the fact that many different functions have the same derivative. For example, integrating \(y'=2x\) gives \(y=x^2+c\text{,}\) where \(c\) is the constant of integration.
To isolate the derivative is to rewrite a DE so the derivative term is alone on one side of the equation. For example, \(y'+18xy=6x\) becomes \(y'=6x-18xy\text{.}\)
The separation of variables method is a procedure for solving a separable DE where we verify the equation is separable, separate the variables onto opposite sides, integrate both sides, and solve for the dependent variable if possible. For example,
An implicit solution is a general solution or particular solution written as a relation involving the dependent variable, rather than with the dependent variable solved for explicitly. For example, \(y^2=2\sin(x)+c\) is implicit because \(y\) has not yet been isolated.
A complete product rule is a two-term expression that matches the pattern \(f(x)g'(x)+f'(x)g(x)\text{,}\) so it can be grouped as a single derivative using the reversed product rule. For example,
An integrating factor is a nonzero function multiplied onto a standard form equation so that the left-hand side becomes a complete product rule. For \(y' + P(x)y = Q(x)\text{,}\) the integrating factor is
A simplified diagram for an autonomous differential equation that shows only the \(y\)-axis, marks equilibria, and uses arrows to show whether solutions move up or down in each interval.
A solution (to a differential equation) written as a formula that gives the exact value of the dependent variable for each input in an interval. For example,
\begin{equation*}
y(t)=e^t
\end{equation*}
is the analytic solution of the IVP\(y'=y,\ y(0)=1\text{.}\)
One repeated application of the basic step in a numerical method. In Euler’s method, each iteration uses the current point to compute the next approximation point.
In differential equation work, like terms are expressions with the same variable or function part that differ only by a constant coefficient. They can be combined by addition or subtraction.
A linear differential equation has constant coefficients if the coefficients of \(y\) and its derivatives are constants rather than functions of the independent variable.
The polynomial equation obtained from an LHCC equation by assuming an exponential solution, which reduces the differential equation to a polynomial in \(r\text{.}\)
A set of solutions is linearly dependent on an interval if there exist constants (not all zero) such that a linear combination of the solutions equals zero.
A set of solutions is linearly independent on an interval if the only linear combination that equals zero is the one with all coefficients equal to zero.
A repeated root is a root of the characteristic equation that appears more than once, indicated by an exponent greater than \(1\) in the factored form of the characteristic polynomial.
For a polynomial with integer coefficients, any rational root \(\frac{p}{q}\) in lowest terms must have \(p\) dividing the constant term and \(q\) dividing the leading coefficient. This helps narrow the possible rational roots of a characteristic polynomial.
The initially unknown constants in a guessed particular solution form that are determined by substituting into the differential equation and matching coefficients of like terms. For example, when the forcing function is a quadratic polynomial:
\begin{equation*}
y_p = Ax^2 + Bx + C
\end{equation*}
If a term in the initial guess for \(y_p\) also appears in the homogeneous solution, multiply any overlapping term in \(y_p\) by the smallest power of \(x\) that eliminates the overlap. For example, if \(e^{3x}\) already appears in \(y_h\text{:}\)
The process of moving a derivative from one function to another inside an integral using integration by parts. In Laplace work, this usually moves the derivative off the unknown function:
In the Laplace transform method, the forward transform means applying the Laplace transform to both sides of a differential equation so it becomes an algebraic equation in the \(s\)-domain.
In the Laplace transform method, the backward transform means applying the inverse Laplace transform term-by-term to \(Y(s)\) to return to the original solution \(y(t)\text{.}\)
A three-step method for solving an initial-value problem: use the forward transform, solve and prepare \(Y(s)\) in the \(s\)-domain, then use the backward transform to recover \(y(t)\text{.}\)
A collection of differential equations that must be solved together because the dependent variables influence one another or are linked in a single model.
A system whose dependent variables appear only to the first power and are not multiplied together or placed inside nonlinear functions. For two variables, a constant-coefficient linear system has the form
For a linear system, the matrix that collects the coefficients of the dependent variables in the equation \(\frac{dY}{dt}=AY\text{,}\) where \(Y\) is the state vector.
\begin{align*}
\frac{dY}{dt} = AY,\qquad A=\begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}
\end{align*}
A system whose highest derivatives are first derivatives. A second-order equation such as \(y''+3y'+2y=0\) can be rewritten as a first-order system by introducing a new variable for the first derivative.
The coordinate plane whose axes are the dependent variables of a two-dimensional system. A solution appears as a path in the \((x,y)\)-plane rather than as separate graphs versus time.
