Question

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

Answer 1

Hint: Degree is the highest power of the highest order derivative involved in a given differential equation.

Complete step-by-step solution -
Now, to study the degree of a differential equation, the key point is that the differential equation must be a polynomial equation in derivatives, i.e. y’, y’’, y’’’ etc. By degree, we mean that the highest power of the highest order derivative in the differential equation given in the question. For example, in the differential equation,
$\dfrac{{{d^3}y}}{{d{x^3}}} + \dfrac{{{d^2}y}}{{d{x^2}}} + y = 0$, the degree is 1, because the power of highest order derivative i.e. $\dfrac{{{d^3}y}}{{d{x^3}}}$ is 1.
Also, the derivatives which are defined as a function, have undefined degree. For example, in the differential equation,
$\dfrac{{dy}}{{dx}} + \sin \left( {\dfrac{{dy}}{{dx}}} \right) = 0$, the degree is undefined, because the highest derivative $\dfrac{{dy}}{{dx}}$ is the function of sin.
So, given differential equation is,
$\dfrac{{{d^2}y}}{{d{x^2}}} + x{\left( {\dfrac{{dy}}{{dx}}} \right)^2} = 2{x^2}\log \left( {\dfrac{{{d^2}y}}{{d{x^2}}}} \right)$.
Now, in this differential equation, the highest order derivative is $\dfrac{{{d^2}y}}{{d{x^2}}}$. But, $\dfrac{{{d^2}y}}{{d{x^2}}}$ is the function of log, so, the degree of the above differential equation is undefined.

Note: For finding the degree of the differential equation, it is important to know what is the highest order derivative in the equation. By order, we mean the maximum times of differentiation, for example $\dfrac{{{d^2}y}}{{d{x^2}}}$ is the second order derivative, because it is differentiated two times. Similarly, $\dfrac{{{d^3}y}}{{d{x^3}}}$ has an order 3 since it is differentiated 3 times. Students generally commit a mistake while finding the degree of the differential equation like this question. In this question, the highest order derivative is the function of log and hence its degree is undefined.