Question

Form a differential equation by eliminating the arbitrary constant $a$ from the equation $y=a\sin \left( 2x+3 \right)$ ?

Answer 1

Hint: Here we have to form a differential equation by the equation given.Firstly we will differentiate the original equation given with respect to $x$. Then we will divide the original equation by the equation obtained after the differentiation. Finally we will cancel out the arbitrary constant and get our desired answer.

Complete step by step answer:
We have been given the equation whose differential equation is to be formed as follows:
$y=a\sin \left( 2x+3 \right)$…..$\left( 1 \right)$
Now for removing the arbitrary constant we will differentiate the above equation as follows:
$\dfrac{d}{dx}y=\dfrac{d}{dx}a\sin \left( 2x+3 \right)$
Now as we know that differentiate of sine is cosine which is define as $\dfrac{d\left( \sin ax \right)}{dx}=\cos ax\dfrac{d\left( ax \right)}{dx}$ using it above we get,
$\Rightarrow \dfrac{dy}{dx}=a\cos \left( 2x+3 \right)\dfrac{d\left( 2x+3 \right)}{dx}$
$\Rightarrow \dfrac{dy}{dx}=a\cos \left( 2x+3 \right)\times 2$

So we get,
$\Rightarrow \dfrac{dy}{dx}=2a\cos \left( 2x+3 \right)$…..$\left( 2 \right)$
On dividing equation (1) by equation (2) we get,
$\Rightarrow \dfrac{y}{\dfrac{dy}{dx}}=\dfrac{a\sin \left( 2x+3 \right)}{2a\cos \left( 2x+3 \right)}$
On cross multiplying we get,
$\Rightarrow y\times \dfrac{2a\cos \left( 2x+3 \right)}{a\sin \left( 2x+3 \right)}=\dfrac{dy}{dx}$
$\Rightarrow \dfrac{dy}{dx}=y\times \dfrac{2\cos \left( 2x+3 \right)}{\sin \left( 2x+3 \right)}$
Now as know $\cot A=\dfrac{\cos A}{\sin A}$using it above we get,
$\therefore \dfrac{dy}{dx}=2y\cot \left( 2x+3 \right)$
So we got the answer as $\dfrac{dy}{dx}=2y\cot \left( 2x+3 \right)$ .

Hence differential equation of $y=a\sin \left( 2x+3 \right)$ by removing the arbitrary constant is $\dfrac{dy}{dx}=2y\cot \left( 2x+3 \right)$.

Note: Differential equations are those equations which have one or more than one derivative of a function. The function or variable whose derivative is present in the equation is known as the dependent variable and the derivative with respect to the variable is known as the independent variable. Arbitrary constants are those constants which can be assigned different values and there is no change in it when the values of the variables are changed in the equation. As the arbitrary constant in the equation is in product with the variable, removing it by differentiation is not possible therefore we have used the division method to remove it.