--- title: "Differential Equations" description: "22 topics in Mathematics, in the order they build on each other." canonical: https://lightmysky.com/learn/mathematics/areas/differential-equations source: https://lightmysky.com/learn/mathematics/areas/differential-equations.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Differential Equations 22 topics in Mathematics, in the order they build on each other. Page: https://lightmysky.com/learn/mathematics/areas/differential-equations - [What a Differential Equation Says and What a Solution Is](https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o): An equation relating a function to its own derivatives constrains a whole family of functions. A general solution carries arbitrary constants; an initial condition picks one member out of the family. - [Direction Fields and Euler's Method](https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR): Draw the slope the equation prescribes at each point and the solution curves appear without any formula. Stepping along those slopes numerically is Euler's method, with error you can reason about. - [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ): When the equation factors into a part in y and a part in x, separate the variables and integrate both sides. The constant of integration is where the initial condition enters. - [First-Order Linear Equations and the Integrating Factor](https://lightmysky.com/learn/mathematics/first-order-linear-equations-and-the-integrating-factor-mt_BXmt2pbWp8): Multiply through by a factor chosen to turn the left side into the derivative of a product, then integrate once. The factor is built from the coefficient of y. - [The Logistic Equation and Saturating Growth](https://lightmysky.com/learn/mathematics/the-logistic-equation-and-saturating-growth-mt_7Y_48Ga33L): Growth proportional to the population and to the room left produces an S-shaped curve with a carrying capacity. The model is where equilibrium solutions first appear. - [Second-Order Linear Equations with Constant Coefficients](https://lightmysky.com/learn/mathematics/second-order-linear-equations-with-constant-coefficients-mt_G7sMq1EHwQ): Try an exponential and the equation becomes a quadratic in the exponent. Real, repeated and complex roots give three shapes of general solution. - [Nonhomogeneous Equations and Particular Solutions](https://lightmysky.com/learn/mathematics/nonhomogeneous-equations-and-particular-solutions-mt_gbkQZte_JZ): The full solution is the homogeneous solution plus any particular one. Undetermined coefficients guesses the shape of the forcing; variation of parameters handles the rest. - [Damped and Driven Oscillations](https://lightmysky.com/learn/mathematics/damped-and-driven-oscillations-mt_Y8K1njb1PH): The same equation governs a mass on a spring and a series circuit. Damping decides how the free motion dies away, and driving near the natural frequency produces resonance. - [Fourier Series](https://lightmysky.com/learn/mathematics/fourier-series-mt_DJ3iwfj7NK): Any reasonable periodic function is a sum of sines and cosines. The coefficients are recovered by integrating against each one, which is what lets a system driven by a complicated periodic input be solved one frequency at a time. - [Series Solutions About an Ordinary Point](https://lightmysky.com/learn/mathematics/series-solutions-about-an-ordinary-point-mt_E6NiSd7MED): When the coefficients are not constant, assume a power series, match coefficients and generate a recurrence. Several named functions of physics are defined this way and no other. - [The Laplace Transform](https://lightmysky.com/learn/mathematics/the-laplace-transform-mt_8j8GXzITN0): An improper integral converts a function of time into a function of a new variable, turning differentiation into multiplication. Its usefulness rests on the transform being reversible. - [Solving Initial Value Problems with Laplace Transforms](https://lightmysky.com/learn/mathematics/solving-initial-value-problems-with-laplace-transforms-mt_0BhjPVNUwl): Transform the equation, solve the resulting algebra for the transform of the unknown, then invert. Initial conditions enter at the start rather than at the end, which is what makes the method suit discontinuous forcing. - [Sturm-Liouville Problems and Eigenfunction Expansions](https://lightmysky.com/learn/mathematics/sturm-liouville-problems-and-eigenfunction-expansions-mt_oqadAPaSsW): Putting a second-order boundary value problem in self-adjoint form makes its eigenfunctions orthogonal, which is why a solution can be written as a series in them at all. Fourier series is then one case rather than a trick that happened to work. - [Green's Functions for Boundary Value Problems](https://lightmysky.com/learn/mathematics/greens-functions-for-boundary-value-problems-mt_2ctFrfKIgW): Solve the problem once for a point source and the solution for any source is an integral against that answer. The Green's function carries the boundary conditions, so it is built from two homogeneous solutions chosen to die at the right end. - [Systems of Differential Equations and Eigenvalue Solutions](https://lightmysky.com/learn/mathematics/systems-of-differential-equations-and-eigenvalue-solutions-mt_BbOqJ4S1ur): Several coupled unknowns become one vector equation. The eigenvalues of the coefficient matrix give the growth rates and the eigenvectors give the directions along which the system decouples. - [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr): Draw the trajectories of a system in the plane of its unknowns. Equilibria are classified by the eigenvalues, and the same classification says whether nearby solutions return or leave. - [Classifying Second-Order PDEs and What the Type Decides](https://lightmysky.com/learn/mathematics/classifying-second-order-pdes-and-what-the-type-decides-mt_hlQWbd1kzR): The sign pattern of the second-order coefficients sorts equations into elliptic, parabolic and hyperbolic. The type decides which data may be prescribed and whether a solution smooths its data out or carries it along. - [Separation of Variables and the Heat Equation](https://lightmysky.com/learn/mathematics/separation-of-variables-and-the-heat-equation-mt_XnPDL0X1TL): Assume the solution is a product of one-variable functions and the equation splits into ordinary ones. The boundary conditions pick out which modes are allowed, and the initial profile is matched by a series in those modes. - [The Wave Equation and Its Characteristics](https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w): Along two families of lines the wave equation reduces to an ordinary derivative, and the solution reads as two fixed shapes travelling in opposite directions. A discontinuity in the data is carried along, not smoothed away. - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V): A solution of Laplace's equation equals its own average over any small sphere, so it can have no interior peak. Uniqueness and stability for the boundary value problem follow from that one observation. - [Weak Solutions and Test Functions](https://lightmysky.com/learn/mathematics/weak-solutions-and-test-functions-mt_AKlC3Ol70p): Multiply the equation by a smooth test function, integrate by parts, and the derivative moves off the unknown. A function with a corner can then solve an equation that in the classical sense it cannot. - [Numerical Schemes for Evolution Equations](https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB): Replace the derivatives by differences and a differential equation becomes a recursion. Consistency and stability together give convergence, and the stability condition is what caps the time step.