Differential Equation, Order, and Degree

Definition 10.1 (Differential equation). A differential equation is any equation containing at least one derivative of an unknown function — either an ordinary derivative (one independent variable) or a partial derivative (two or more independent variables). For $y=f(x)$ with $y$ dependent and $x$ independent, each of the following is a differential equation: $\dfrac{dy}{dx}=0$; $\dfrac{dy}{dx}=\sin x$; $\dfrac{dy}{dx}+y=x+7$; $\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}+y=\sin x$; $e^x\dfrac{dy}{dx}=\ln x\ (x>0)$; and even $\tan^{-1}\!\left(\dfrac{d^2y}{dx^2}+\dfrac{y}{x^2}\right)=\dfrac{dy}{dx}$.

Definition 10.2 (Order). The order of a differential equation is the order of the highest derivative present. If the highest derivative of $y$ appearing is the $k$th, the order is $k$ — necessarily a positive integer. For example, $\dfrac{d^3y}{dx^3}-5\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}=0$ has order $3$.

Definition 10.3 (Degree). If a differential equation is expressible in polynomial form in its derivatives, the degree is the integral power to which the highest-order derivative is raised. More precisely, once the equation is written in polynomial form so that:

1. none of the derivatives carry fractional powers, and
2. the highest-order derivative is not the argument of a transcendental (trigonometric, exponential, logarithmic, …) function, and the coefficient of the term containing it is only a function of $x$, $y$, or lower-order derivatives — never a transcendental function of a derivative — then the degree is that integral power. If the equation cannot first be put into this polynomial form with the highest-order derivative as its leading term, the degree is not defined (though the order can still be stated). Whenever it exists, the degree is always a positive integer. Worked technique for finding degree (illustrated on several equations of increasing difficulty):

• $3\left(\dfrac{d^2y}{dx^2}\right)=\left[4+\left(\dfrac{dy}{dx}\right)^2\right]^{3/2}$: squaring both sides clears the fractional power, giving $9\left(\dfrac{d^2y}{dx^2}\right)^2=\left[4+\left(\dfrac{dy}{dx}\right)^2\right]^3$ — order $2$, degree $2$.
• $1+\left(\dfrac{dy}{dx}\right)^2=\left(y\dfrac{d^2y}{dx^2}\right)^{2/3}$: cubing clears the fractional power — order $2$, degree $2$.
• $\sin\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}+x=0$: the highest-order derivative $\dfrac{d^2y}{dx^2}$ is not itself inside a transcendental function, but because a derivative appears as the argument of $\sin$ at all, the equation cannot be reduced to polynomial form — order $2$, degree not defined.
• $e^{x}\dfrac{dy}{dx}+\dfrac{d^2y}{dx^2}=\sin x$: again a derivative sits inside a transcendental-function context that blocks polynomial form — order $2$, degree not defined.
• Equations such as $10(y''')^4+7(y'')^5+5(y')+\sin(y)=0$ or $\cos(y'''+y'')+y'=\sin x$ similarly have their order but no defined degree. …