Question

Write the degree of the differential equation $x{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}+y{{\left( \dfrac{dy}{dx} \right)}^{4}}+{{x}^{3}}=0$.

Answer 1

We start solving the problem by recalling the definitions of order and degree of a differential equation. We find the highest order derivative in order to find the value of the order of the differential equation. We then check the power of that highest order derivative to get the required degree of the given differential equation.

Complete step-by-step answer :
According to the problem, we have a differential equation $x{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}+y{{\left( \dfrac{dy}{dx} \right)}^{4}}+{{x}^{3}}=0$ and we need to find the degree of the given differential equation.
Let us recall the definition of order and degree of a differential equation.
We know that the order of a differential equation is defined as the order of the highest order derivative of dependent variable with respect to independent variable which were present in the differential equation.
We have variables x and y involved in the problem and from the derivatives, we can see that x is an independent variable and y is a dependent variable.
So, $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ is the higher order derivative involved in the differential equation.
We know the order of $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ is 2, which makes the order of the given differential equation 2.
We know that degree is defined as the highest power of the highest order derivative involved in the differential equation.
In the differential equation $x{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}+y{{\left( \dfrac{dy}{dx} \right)}^{4}}+{{x}^{3}}=0$, we can see the highest order derivative is $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$. We can see that the power of $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ given in the differential equation is 3. So, this makes the degree of the differential equation is 3.
We have found the order and degree of the differential equation $x{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}+y{{\left( \dfrac{dy}{dx} \right)}^{4}}+{{x}^{3}}=0$ as 2 and 3.
∴ The order and degree of the differential equation $x{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{3}}+y{{\left( \dfrac{dy}{dx} \right)}^{4}}+{{x}^{3}}=0$ is 2 and 3.

Note : We should remember that the order and degree of any differential equation must be integers. We should not tell the degree if all the derivatives of the dependent variable cannot be written as a polynomial. If we get the differential equation containing $\sin \left( \dfrac{dy}{dx} \right)$ etc., we cannot say the degree. We need to make sure that all derivatives involved in differential equations are lying in a polynomial.