## Degree vs Order

This article aim to distinguish between the two concepts: Degree & Order.
Many may ask are degree and order the same concept or not ? In fact, they are both related to differential equation (DE), because differential equations are often classified with respect to order and degree.

Degree of DE: is the power of the highest order derivative in the equation.

Order of DE: is the order of the highest order derivative existed in the equation.

Examples 1:

$$\frac{\partial^2 y}{\partial x^2} -5x\left ( \frac{\partial y}{\partial x} \right )^{4} e^{x} = \frac{\partial y}{\partial x}$$ has order 3 and degree 1.

Examples 2:

$$\left (\frac{\partial^2 y}{\partial x^2} \right ) ^{3} - \frac{\partial y}{\partial x} = cos(x)$$ has order 2 and degree 3.

Examples 3:

$${y}'' - xy = \left ({y}' \right )^{4}$$ has order 2 and degree 1.

Examples 4:

$$x\left ( \frac{\partial^2 y}{\partial x^2} \right )^{3} \frac{\partial y}{\partial x} - 7e^{x}\left ( \frac{\partial^2 y}{\partial x^2} \right )+yln(x) = 5$$ has order 3 and degree 2.

Examples 5:

This Example illustrate the weakness of definition in term of degree .i.e

$$e^{\frac{\partial y}{\partial x}} + cos\left ( x\frac{\partial^2 y}{\partial x^2} \right ) + tan\left ( \frac{\partial^3 y}{\partial x^3} \right ) + 3xy = 0$$ which has order 3 and no degree can be determined.

Thus, from the last example we notice that despite the above definition of order is complete, the degree is not so.