Question

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

Answer 1

We start this question by going through the concept of degree of a differential equation. Then we examine the given differential equation to find the higher order derivative and find the degree of it to find the degree of the given differential equation.

Complete step by step answer:
First let us go through the definition of differential equation.
An equation that involves dependent and independent variables and the derivatives of dependent variables with respective to independent variables is called a differential equation.
Now let us go through the definition of degree of a differential equation.
The degree of a differential equation, which is expressed as a polynomial in the derivatives is the degree of the highest order derivative in the polynomial and the polynomial should be free of any fractional form with derivatives. Then the degree of highest order derivative is the degree of the given differential equation.
The differential equation we were given is ${{\left( \dfrac{dy}{dx} \right)}^{4}}+3x\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$.
As we see the above differential equation is free of any fractional form of derivatives. So, now we can find the degree of the given differential equation.
As we see in ${{\left( \dfrac{dy}{dx} \right)}^{4}}+3x\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ we have two terms with derivatives one is $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ and the other is ${{\left( \dfrac{dy}{dx} \right)}^{4}}$. Among them the higher order derivative is $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$.
The degree of $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ is 1.
So, the degree of the differential equation ${{\left( \dfrac{dy}{dx} \right)}^{4}}+3x\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ is 1.

So, the correct answer is 1.

Note: The major mistake that many people make while solving this question is, they confuse the degree of the differential equation with the order of the differential equation. Order of the differential equation is the order of the highest differential coefficient in the equation. In our question ${{\left( \dfrac{dy}{dx} \right)}^{4}}+3x\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$, highest differential is $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$. So, one might write the degree as 2. So, it is necessary to remember the difference between order and degree of a differential equation.