Summary

Differential equation. An equation relating an independent variable, a dependent variable, and derivatives of the dependent variable; this chapter treats only ordinary differential equations (one independent variable).

Order and degree. Order $=$ order of the highest derivative present. Degree $=$ power of that highest derivative, once the equation is polynomial in derivatives (clear radicals/fractional powers first); degree is not defined if a derivative sits inside a transcendental function.

Solutions. $y=\phi(x)$ is a solution if substituting it (and its derivatives) satisfies the equation identically. The general solution of an order-$n$ equation carries $n$ independent arbitrary constants; a particular solution fixes those constants, typically via an initial condition $y(x_0)=y_0$ (an initial value problem).

Separation of variables. $\dfrac{dy}{dx}=f(x)g(y) \implies \displaystyle\int\frac{dy}{g(y)} = \int f(x)\,dx + C$.

Homogeneous equations. $\dfrac{dy}{dx}=F(y/x)$: substitute $y=vx$, $\dfrac{dy}{dx}=v+x\dfrac{dv}{dx}$, reducing to the separable equation $x\dfrac{dv}{dx}=F(v)-v$; back-substitute $v=y/x$ at the end. (Mirror: $x=vy$ when the equation is a pure function of $x/y$.)

Linear equations, $y$-form. $\dfrac{dy}{dx}+Py=Q$ ($P,Q$ functions of $x$): $\text{I.F.}=e^{\int P\,dx}$, general solution $y\cdot\text{I.F.} = \displaystyle\int Q\cdot\text{I.F.}\,dx + C$. …