Question

The order and degree of the differential equation \[{{\left( {{y}^{'''}} \right)}^{2}}+{{\left( {{y}^{''}} \right)}^{3}}-{{\left( {{y}^{'}} \right)}^{4}}+{{y}^{5}}=0\text{ is:}\]
(a) 3 and 2
(b) 1 and 2
(c) 2 and 3
(d) 1 and 4
(e) 3 and 5

Answer 1

Hint: First look at the definition of the order and degree of a differential equation. Then apply these definitions to the given equation. By the definition of the order, you can find the value of the order. By definition of the degree, you can find the value of the degree. The values of the order and degree are the required result.

Complete step-by-step answer:
Order of a differential equation: Differential equations are classified on the basis of the order. The order of a differential equation is the order of the highest derivative (also known as differential coefficient) present in the equation. In the equation, compare all the power of the derivatives.
Given the differential equation in the question is written as
\[{{\left( {{y}^{'''}} \right)}^{2}}+{{\left( {{y}^{''}} \right)}^{3}}-{{\left( {{y}^{'}} \right)}^{4}}+{{y}^{5}}=0\]
The power of the function y in the question is given as 5.
The power of the first derivative of y in the expression is 4.
The power of the second derivative of y in the expression is 3.
The power of the third derivative of y in the expression is 2.
The highest derivative of y in the question is the third derivative which is y’’’.
The order of this term can be found as follows:
\[{{y}^{'''}}=\dfrac{{{d}^{3}}y}{d{{x}^{3}}}\]
So, the order of the third derivative is 3. So, the order of the given expression in the question is 3.
Degree of a differential equation: The degree of a differential equation is the power of its highest derivative after the equation has been made rational and integral in all of its derivatives.
From the statements written above, we can see that the highest derivative from all the derivatives present is the third derivative.
The power of the third derivative is written as 2 in the question. So, this is the power of the highest derivative. So, we can say that the degree of the given expression has a value of 2.
Order and degree of the given differential equation are 3 and 2.
Hence, option (a) is the right answer.

Note: Generally, students take the definitions in reverse, that is they apply the definition of the degree to find the value of the order. By this, you get the wrong answer. Some misconceptions are believed such as maximum power is the degree, but it is wrong. Only the power of the highest derivative is the degree of the differential equation.