Answer

Verified

427.5k+ views

**Hint**: In this question, we have to derive the differential equation from the given expression. There are two variables given in the expression $ x $ and $ y $ . According to the question, the differential equation has to be a second order differential equation in terms of $ y $ which means that we have to differentiate the given expression two times w.r.t. $ x $ .

We know that $ \dfrac{{dy}}{{dx}} = {y_1} $

$ \dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) = {y_2} $

And,

$ \dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}} $

**:**

__Complete step-by-step answer__Given: The expression given is-

$ a{x^2} + b{y^2} = 1 $

Where $ a{\text{ and b}} $ are arbitrary constants.

In order to find the differential form of this expression we have to do the derivation. Since there are 2 arbitrary constants in the expression and to remove them, we have to do the differentiation twice.

So, differentiating both sides of the expression w.r.t. $ x $ we get,

$

\dfrac{d}{{dx}}\left( {a{x^2} + b{y^2}} \right) = \dfrac{d}{{dx}}\left( 1 \right)\\

a\dfrac{d}{{dx}}\left( {{x^2}} \right) + b\dfrac{d}{{dx}}\left( {{y^2}} \right) = 0

$

Solving this we get,

$ a \times 2x + b \times 2y \times \dfrac{{dy}}{{dx}} = 0 $

We can write the first derivative $ \dfrac{{dy}}{{dx}} = {y_1} $ in the equation. So, we get,

$

2ax + 2by{y_1} = 0\\

2\left( {ax + by{y_1}} \right) = 0

$

We can write this as-

$ ax + by{y_1} = 0 $

This is our first equation.

Now differentiating both sides again w.r.t. $ x $ we get,

$

\dfrac{d}{{dx}}\left( {ax + by{y_1}} \right) = 0\\

a\dfrac{d}{{dx}}\left( x \right) + b\dfrac{d}{{dx}}\left( {y{y_1}} \right) = 0

$

We can write

$

\dfrac{d}{{dx}}\left( {y{y_1}} \right) = y\dfrac{d}{{dx}}\left( {{y_1}} \right) + {y_1}\dfrac{d}{{dx}}\left( y \right)\\

= y\dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) + {y_1}\left( {\dfrac{{dy}}{{dx}}} \right)

$

We know that $ \dfrac{{dy}}{{dx}} = {y_1} $ and

$

\dfrac{d}{{dx}}\left( {\dfrac{{dy}}{{dx}}} \right) = \dfrac{{{d^2}y}}{{d{x^2}}}\\

= {y_2}

$

So, $

\dfrac{d}{{dx}}\left( {y{y_1}} \right) = y \times {y_2} + {y_1} \times {y_1}\\

= y{y_2} + {y_1}^2

$

Substituting these values in the equation we get,

$

a \times 1 + b \times \left( {y{y_2} + {y_1}^2} \right) = 0\\

a + b\left( {y{y_2} + {y_1}^2} \right) = 0\\

a = - b\left( {y{y_2} + {y_1}^2} \right)

$

Putting this value of $ a $ in the first equation we get,

$

- b\left( {y{y_2} + {y_1}^2} \right)x + by{y_1} = 0\\

y{y_1} = \left( {y{y_2} + {y_1}^2} \right)x\\

y{y_1} = xy{y_2} + x{y_1}^2

$

We can write this equation as-

$ xy{y_2} + x{y_1}^2 - y{y_1} = 0 $

Therefore, the differential equation is $ xy{y_2} + x{y_1}^2 - y{y_1} = 0 $

**So, the correct answer is “Option A”.**

**Note**: The number of times we have to do the differentiation depends upon the numbers of arbitrary constants present in the expression. In the second-order differential equation, the highest order of the equation is 2. The order of the derivative is defined as the number of times of the derivation of the variable.

For example, the order of the differential $ \dfrac{{{d^3}y}}{{d{x^3}}} $ is 3.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

How do you graph the function fx 4x class 9 maths CBSE

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE