Order, Degree and Solutions

The two classifiers:

• Order = the order of the highest derivative present (e.g. $d^2y/dx^2$ gives order 2).
• Degree = the power of the highest-order derivative AFTER the equation is made free of radicals and fractional powers (made polynomial in the derivatives).
• Degree is undefined when a derivative appears inside a transcendental function, e.g. $\cos\!\big(\tfrac{dy}{dx}\big)$ or $\ln\!\big(\tfrac{dy}{dx}\big)$.
• Tip: $\dfrac{dx}{dy} = \Big(\dfrac{dy}{dx}\Big)^{-1}$ — rewrite mixed derivatives in one form before reading the degree.
• An integral sign in the equation: differentiate both sides first. $k\dfrac{dy}{dx} = \displaystyle\int\Big[1+\Big(\dfrac{dy}{dx}\Big)^2\Big]^{2/3}dx$ is not yet a differential equation you can classify. Differentiating removes the integral and raises the order by one: $k\dfrac{d^2y}{dx^2} = \Big[1+\Big(\dfrac{dy}{dx}\Big)^2\Big]^{2/3}$. Only now clear the fractional power and read order and degree.

Solutions and what they tell you:

• A general solution of an order-$n$ ODE contains exactly $n$ arbitrary constants; a particular solution fixes them via conditions.
• So the order = number of independent arbitrary constants in the family. $y=a\cos x+b\sin x$ (two constants) → order 2.
• An ODE like $\dfrac{d^2y}{dx^2}+k^2y=0$ has periodic (SHM) solutions; $\dfrac{d^2y}{dx^2}-k^2y=0$ gives exponential growth.
• Two members of one solution family never meet when they differ by an additive constant: $y=x^2+c_1$ and $y=x^2+c_2$ are vertical translates of the same curve, so setting them equal gives $c_1=c_2$, a contradiction — "how many points of intersection?" is none. (Families where the constant multiplies, like $y=cx^2$, DO share the origin.)