Question

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

Answer 1

Hint: We will find the degree of the given differential equation by finding its order first. We know that the order of the differential equation is the order of highest derivative also known as differential coefficient present in the equation. We can also define the degree of differential equations as the power of the highest derivative, after the equation has been made rational and integral in its entire derivative.

Complete step by step answer:
We have to find the degree of the given differential equation.
Given differential equation is ${{x}^{3}}{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}+x{{\left( \dfrac{dy}{dx} \right)}^{4}}=0$
For that first we have to find the order of the given differential equation.
As we know, differential equations are classified on the basis of the order of differential equations.
Order of the differential equation is the order of highest derivative also known as differential coefficient present in the equation.
If we observe in the given differential equation, the order of the highest derivative is 2.
Hence order of the given differential equations is 2.
Now we will find the degree of the given differential equations.
Degree of differential equations is the power of the highest derivative, after the equation has been made rational and integral in its entire derivative.
So in our given differential equations ${{x}^{3}}{{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}+x{{\left( \dfrac{dy}{dx} \right)}^{4}}=0$
Power of the highest derivative is 2 i.e. we have ${{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}$

So from that we can say that the degree of the given differential equation is 2.

Note: Remember the degree of the differential equation can be found when it is in the polynomial form otherwise the degree cannot be defined. And the degree of differential equations is the power of the highest derivative, after the equation has been made rational and integral in its entire derivative. A mistake that can be made here is that students can get concepts of order and degree wrong and write 4 as the answer.