The dependent variable represents the unknown function that you are solving for and always has derivatives applied to it. The dependent variable is a function of the independent variable. In
Parts of an equation separated by \(+\text{,}\)\(-\text{,}\) or \(=\) and each containing a different form of the dependent variable. The collection of all terms without a dependent variable is referred to as a free term.
Every first-order differential equation can be written in the form
\begin{equation*}
f(x, y, y') = 0\text{.}
\end{equation*}
This just means that all the terms in the equation have been moved to the left-hand side, which can only contain the independent variable, \(x\text{,}\) the dependent variable, \(y\text{,}\) and its derivative, \(y'\text{.}\)
A function, \(\mu\text{,}\) multiplied onto the standard form, above, to reverse the product rule, leading to the equation, \(\mu' = P\mu\text{,}\) with the solution
A picture of the slopes that a solution curve must follow at each point of the plane, drawn as short segments. It represents the entire family of solutions to \(y' = f(t, y)\text{;}\) here \(f(t, y)\) is the “slope generator” that outputs the required slope at any point \((t, y)\text{.}\)
A compressed, one-dimensional summary of an autonomous slope field drawn on a vertical \(y\)-axis: equilibria are marked with solid dots, regions where \(f(y) > 0\) get upward arrows, and regions where \(f(y) < 0\) get downward arrows.
A qualitative change in a system’s dynamics — the number or stability of equilibria changes — as a parameter passes a critical value. For example, \(\frac{dx}{dt} = \mu - x^2\) has two equilibria \(x = \pm\sqrt{\mu}\) for \(\mu > 0\text{,}\) one at \(\mu = 0\text{,}\) and none for \(\mu < 0\) — a saddle-node bifurcation at \(\mu = 0\text{.}\)
with equilibria \(P = 0\) (extinction, unstable) and \(P = K\) (stable). The constant \(K\) is the carrying capacity — the largest population the environment can sustain.
An exact solution expressed as a formula, such as \(y(t) = e^{-3t}\sin(2t)\text{,}\) into which any value of \(t\) can be substituted to obtain the exact value of \(y\text{.}\)
A procedure used when a differential equation has no tidy closed-form solution. Rather than handing you \(y(t)\) as a formula, it builds an approximation one step at a time, starting from what you know and using the differential equation to predict what happens next.
The result of a numerical method: a list of values approximating \(y(t)\) at specific times — a table of \(t\)-values and approximate \(y\)-values (points, not a curve). These approximations carry small errors in exchange for handling equations that analytic methods cannot.
The general solution to a non-homogeneous equation is the sum of the general solution to the corresponding homogeneous equation and a particular solution.
The general solution of an LHCC equation is a linear combination of the independent solutions coming from the characteristic roots. Distinct real roots \(r_1, \ldots, r_n\) give
A repeated root of multiplicity \(m\) contributes an extra factor of \(x\) (up to \(x^{m-1}\)), and a complex pair \(r = \alpha \pm i\beta\) contributes \(e^{\alpha x}(c_1\cos(\beta x) + c_2\sin(\beta x))\text{.}\)
A particular solution, \(y_p\text{,}\) is a function that can be plugged into (C.3) and all the terms combine to give the forcing function, \(f(x)\text{.}\)
A homogeneous solution, \(y_h\text{,}\) is a function such that, when it is plugged into (C.3) all the terms cancel to zero. That is, \(y_h\) is the solution to the homogeneous version of (C.3) (\(f(x) = 0 \)).
Figure383.Comparison of a homogeneous equation (left) and a nonhomogeneous equation (right). In both, the solutions must simplify in a specific way when substituted into the equation.
In this section, we introduced the concept of the forward Laplace transform and derived some common Laplace transforms that we will use throughout this chapter. The following points summarize the essential concepts from the forward Laplace transform section:
Differential \(\to\) Algebraic Equations. The Laplace transform converts a differential equation into an algebraic equation, simplifying the solution process by eliminating derivatives.
Laplace Transform Concept. Applying the Laplace transform to a differential equation involves transforming each term by multiplying by \(e^{-st}\) and integrating with respect to \(t\) from \(0\) to \(\infty\text{,}\) but is often simplified by directly applying the Laplace operator, \(\laplacesym\text{.}\)
Linearity Property. The Laplace transform is linear, meaning it distributes across addition and subtraction, and allows for constants to be factored out. This property is essential for transforming complex equations.
Transforming Initial Conditions. Initial conditions are incorporated directly into the Laplace-transformed equation, modifying the transformed terms to include initial values, making it easier to solve the resulting algebraic equation.
Common Function Transforms. The Laplace transforms of common functions, such as exponentials, sines, cosines, and polynomials, are essential tools in transforming differential equations and are summarized in the provided table.
Transforming Derivatives. The Laplace transform of a derivative, \(y'(t)\) or higher, transfers the derivative onto the Laplace variable \(s\text{,}\) reducing the order of the equation while introducing initial condition terms.
Multiplication by \(e^{at}\) and \(t^n\). When multiplying a function by an exponential \(e^{at}\text{,}\) the Laplace transform shifts by \(a\) in the \(s\)-domain, and multiplying by \(t^n\) corresponds to differentiating the transform \(n\) times with respect to \(s\text{,}\) introducing a sign change.
Transforming the Entire Equation. The process of applying the Laplace transform to an entire differential equation with initial conditions involves systematically transforming each term and leads to a simplified algebraic equation in the \(s\)-domain, ready for solving.
In this section, we introduced the concept of the forward Laplace transform and derived some common Laplace transforms that we will use throughout this chapter. The following points summarize the essential concepts from the forward Laplace transform section:
Common Forms: A table of common Laplace transforms is provided, which doubles as a reference for inverse transforms. The focus is on recognizing forms that match the table entries for functions like \(\sin(bt), \cos(bt)\text{,}\) and others.
Direct Computation: When the function of \(s\) directly matches a form in the common Laplace transform table, the inverse Laplace transform can be easily computed.
Modifying Functions: When a function doesn’t match a known form, minor modifications, such as multiplying by missing constants or splitting fractions, can help.
Completing the Square: When dealing with quadratic expressions in the denominator, especially when the discriminant is negative, completing the square can transform the expression into a form that matches known inverse Laplace transforms. Several examples demonstrate this technique.
Partial Fraction Decomposition: For more complex rational functions, partial fraction decomposition breaks down the function into simpler fractions that match the common transform forms.
Completing the square is an essential technique for transforming quadratic expressions that don’t directly match a form in the table of common Laplace transforms. However, it’s not the only strategy available. In this section, we’ll explore another important technique: partial fraction decomposition. This method is useful for breaking down complex fractions into simpler components that can each be matched with forms in the Laplace transform table.
When we want to take the inverse Laplace transform of a rational function with a second-degree polynomial in the denominator, we may complete the square or we may do a partial fraction decomposition. How will we know which is appropriate? Here are a few guidelines for you to consider.
Does the denominator factor in an obvious way? If so, factor the denominator and do a partial fraction decomposition if necessary.
If instead you end up with subtraction outside the parentheses, as in \((s - a)^2 - b^2,\) then you should factor and do a partial fraction decomposition. You may consider using the quadratic formula if the factorization is not obvious to you.
A function built from different parts over specific regions of its domain. Each piece has its own behavior, but together they create the entire function.
Multiply each piece by the switch for its active interval, then add the results. For a three-part function with cut points \(c \lt d\text{:}\)
\begin{align*}
P(t) \text{ on } t \lt c \amp\ \longrightarrow\ P(t)\,\big(1 - u_c(t)\big),\\
Q(t) \text{ on } c \le t \lt d \amp\ \longrightarrow\ Q(t)\,\big(u_c(t) - u_d(t)\big),\\
R(t) \text{ on } t \ge d \amp\ \longrightarrow\ R(t)\,u_d(t).
\end{align*}
Solve algebraically, then prepare each part as \(F(s)\,e^{-cs}\) — collect the exponential factors, partial-fraction decompose \(F(s)\text{,}\) and precompute \(f(t) = \ilap{F(s)}\text{.}\)
A collection of differential equations that must be solved together because they track several unknowns that may influence each other. A two-variable first-order system has the form
\begin{align*}
\frac{dx}{dt} \amp= a x + b y,\\
\frac{dy}{dt} \amp= c x + d y.
\end{align*}
A system is uncoupled when neither equation contains the other’s variable, partially coupled when the interaction runs one way, and fully coupled when both variables affect each other so that neither equation stands alone.
A system is linear if every equation is linear in its dependent variables — the unknowns and their derivatives appear without products, powers, or nonlinear functions. The dimension of the system is the number of dependent variables.
Collecting the constants into the coefficient matrix \(A\) and the unknowns into the state vector \(\vec{X}\) writes the whole system as one compact equation:
\begin{equation*}
\frac{d\vec{X}}{dt} = A\vec{X}, \qquad
A = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}, \qquad
\vec{X} = \begin{bmatrix} x \\ y \end{bmatrix}.
\end{equation*}
Any higher-order equation becomes a first-order system by naming the derivatives as new variables. For \(y'' + 3y' + 2y = 0\text{,}\) let \(u = y\) and \(v = y'\text{:}\)
For a complex pair \(r = a \pm bi\text{,}\) the real part \(a\) sets a growth factor \(e^{at}\) and the imaginary part \(b\) produces rotation through \(\cos(bt)\) and \(\sin(bt)\text{.}\)
A solution traces a trajectory through the phase plane (the \(xy\)-plane); the collection of all trajectories is the phase portrait. For a \(2 \times 2\) system with distinct eigenvalues, the eigenvalues classify the equilibrium at the origin:
