Differential Equations.Simplified.

An equation containing derivatives is called a differential equation. We classify DEs by order, degree, and type (ODE vs PDE), and learn to form DEs by eliminating arbitrary constants.

If you can rearrange a first-order DE so that all y terms (including dy) are on one side and all x terms (including dx) are on the other, just integrate both sides.

A DE dy/dx = f(x,y)/g(x,y) is homogeneous if f and g are homogeneous functions of the same degree. Substitute y = vx to convert it into a separable equation.

Equations like dy/dx = (ax+by+c)/(Ax+By+C) look almost homogeneous, but constants c and C spoil things. A coordinate shift or substitution fixes this.

A first-order linear DE has the form dy/dx + P(x)y = Q(x). Multiply by the integrating factor e^(integral P dx) and the left side becomes an exact derivative.

A Bernoulli equation has the form dy/dx + Py = Qy^n. Divide by y^n, substitute z = y^(1-n) to convert it into a linear equation.

M dx + N dy = 0 is exact if dM/dy = dN/dx. The solution: integrate M w.r.t. x (y constant) plus the y-only terms of N integrated w.r.t. y.

For ay'' + by' + cy = R(x), the complete solution is y = C.F. + P.I. The C.F. comes from the auxiliary equation; the P.I. depends on the form of R(x).

A Cauchy-Euler equation has variable coefficients as powers of x. Substitute x = e^z to convert it into a constant-coefficient equation.

Works for ANY second-order linear DE. Especially useful when R(x) is sec x, tan x, cosec x, or log x where standard operator methods fail.

When two dependent variables are functions of a single independent variable with linked derivatives, eliminate one variable to get a single ODE, solve, then back-substitute.

DEs model real-world phenomena: electrical circuits, Newton's cooling, orthogonal trajectories, population growth/decay, spring-mass systems, and more.