Question

Find the order and degree of $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{\dfrac{3}{2}}}$.
A) $\left( 1,2 \right)$
B) $\left( 2,2 \right)$
C) $\left( 2,\dfrac{3}{2} \right)$
D) $\left( \dfrac{3}{2},2 \right)$

Answer 1

Hint: The above given equation is a differential equation. In mathematics, a differential equation is an equation that contains one or more functions with its derivatives. The derivatives of function define the rate of change of a function at a point. Here we have to find out the degree and order of the given differential equation. Now the order of a differential equation depends on the derivative of the highest order in the equation. And the degree of any differential equation is determined by the highest exponent on any variables involved.

Complete step by step solution:
The given differential equation is:
$\Rightarrow \dfrac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{\dfrac{3}{2}}}$
Since, to get the value of degree and order of the given differential equation, first we have to remove the fraction power from the above equation.
For this we will simplify the given differential equation.
Now,
$\Rightarrow \dfrac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{\dfrac{3}{2}}}$
First we will have to apply square on both sides of the given differential equation, then we get
\[\begin{align}
  & \Rightarrow \dfrac{{{d}^{2}}y}{d{{x}^{2}}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{\dfrac{3}{2}}} \\
 & \Rightarrow {{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{\dfrac{3}{2}\times 2}} \\
\end{align}\]
Now we can see in the right side of the equation we can easily cancel out the numerator and denominator $2$ from the above equation, then we get
$\Rightarrow {{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}}={{\left[ 1+{{\left( \dfrac{dy}{dx} \right)}^{2}} \right]}^{3}}$
Therefore, we get the simplest form of the given differential equation.
Now the order of the given differential equation is the derivative of the highest order, so we can see $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ is the highest order derivative, so order of the differential equation is $2$.
The degree of the differential equation is the power of the highest order derivative. Here the power of highest order derivative is also $2$ $\left( {{\left( \dfrac{{{d}^{2}}y}{d{{x}^{2}}} \right)}^{2}} \right)$ .

So, the correct answer is “Option B”.

Note: In order to find the order and degree of any differential equation we can go wrong by not simplifying the given equation if we have any power in the fraction form. We cannot write the degree of any differential equation in a fraction. So first we have to simplify the given differential equation if necessary.