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.