Question

Find the order and degree of the differential equation ${{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{6}}}{{\left( \dfrac{dy}{dx} \right)}^{\dfrac{1}{3}}}=5$?

Answer 1

We start solving the problem by recalling the definitions of order and degree of the differential equation. We then find the highest order derivative present in the given differential equation to find the order of the differential equation. We then make the necessary arrangements to eliminate the fractional powers present in the given differential equation. We then check whether all the given derivatives can be written in a polynomial and check the degree of the highest derivative to get the degree of the differential equation.

Complete step-by-step answer:
According to the problem, we need to find the order and degree of the differential equation ${{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{6}}}{{\left( \dfrac{dy}{dx} \right)}^{\dfrac{1}{3}}}=5$ ---(1).
Let us first recall the definitions of order and degree of the differential equation.
Order: We know that the order of a differential equation is defined as the highest order of the derivative involved in the differential equation.
Degree: We know that the degree of a differential equation is defined as the positive degree (should be integer) of the highest order derivative present in the differential equation when all the derivatives present in it are free from fractional powers and should be in polynomial.
From equation (1), we can see that the order of the highest derivative present in the differential equation is 3. This tells us that the order of the given differential equation is 3.
Now, we can see that the derivatives present in the differential equations have fractional powers. So, let us first cube on both sides.
So, we get ${{\left( {{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{6}}}{{\left( \dfrac{dy}{dx} \right)}^{\dfrac{1}{3}}} \right)}^{3}}={{5}^{3}}$.
$\Rightarrow {{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{2}}}\left( \dfrac{dy}{dx} \right)=125$.
Now, let us square on both sides.
$\Rightarrow {{\left( {{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{2}}}\left( \dfrac{dy}{dx} \right) \right)}^{2}}={{125}^{2}}$.
$\Rightarrow \left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right){{\left( \dfrac{dy}{dx} \right)}^{2}}=15625$.
We can see that the highest order derivative of this differential equation has degree 1. So, this makes the degree of the given differential equation as 1.
So, we have found the order and degree of the differential equation ${{\left( \dfrac{{{d}^{3}}y}{d{{x}^{3}}} \right)}^{\dfrac{1}{6}}}{{\left( \dfrac{dy}{dx} \right)}^{\dfrac{1}{3}}}=5$ as 3 and 1.

Note: We should not say that the degree of the given differential equation is 1 as we can see that we can write the derivatives $\dfrac{{{d}^{3}}y}{d{{x}^{3}}}$, ${{\left( \dfrac{dy}{dx} \right)}^{2}}$ in a polynomial with degree as positive number. We should follow all the rules of the differential equation before finding the order and degree of the differential equation. We should remember that we can find order irrespective of the degree of the differential equation which is not the case for finding the degree. Similarly, we can expect problems to find the order and degree of the differential equation which represents the circles with centre at origin.