Differential Equation Modelling

An equation involving derivatives of a dependent variable with respect to an independent variable.

Order = highest derivative present.
Degree = power of that highest derivative (when polynomial in derivatives).

General: contains arbitrary constants. Particular: obtained by assigning specific values.

Eliminate arbitrary constants from a given equation to form the differential equation.

If $\frac{dy}{dx}=f(x)g(y)$ then $\frac{1}{g(y)}dy=f(x)dx$, and integrate both sides.

If $\frac{dy}{dx}=F(y/x)$, put $y=vx$ so $\frac{dy}{dx}=v+x\frac{dv}{dx}$.

Standard form $\frac{dy}{dx}+Py=Q$, IF $=e^{\int P\,dx}$, solution $y\cdot IF=\int Q\cdot IF\,dx+C$.

$\frac{dy}{dx}+Py=Qy^n$. Divide by $y^n$ and convert to linear form.

Given $M\,dx+N\,dy=0$, exact if $\partial M/\partial y=\partial N/\partial x$.

Slope of curve = $dy/dx$; slope of orthogonal trajectory = $-1/(dy/dx)$.

$\int dx=x+C$ · $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ · $\int e^x dx=e^x+C$ · $\int \frac{1}{x}dx=\ln|x|+C$.

Every DE has infinitely many solutions; initial conditions give the particular solution; the integration constant is compulsory.