Need solution for RD Sharma maths class 12 chapter 21 Differential Equation exercise Fill in the blank question 27

Order = 2 or 2nd order

Given:

The order of differential equation representing all circles of radius r is ____

Hint:

Here, Order = Highest derivative

Solution:

Any circle with given radius can be written as (x-h)2+(y-k)2=r2

Where (h, k) be the Centre of the circle and radius is constant.

Differentiating both side w.r.t x we get

\begin{aligned} &2(x-h)+2(y-k)\frac{\mathrm{d} y}{\mathrm{d} x}=0 \\ &\Rightarrow 2(y-k)\frac{\mathrm{d} y}{\mathrm{d} x}=-2(x-h) \\ &\Rightarrow (y-k)\frac{\mathrm{d} y}{\mathrm{d} x}=-(x-h) \\ &\Rightarrow y\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}=-1 \\ &\Rightarrow \frac{\mathrm{d} ^{2}y}{\mathrm{d} x^{2}}+\left ( \frac{\mathrm{d} y}{\mathrm{d} x} \right )^{2}=-\frac{1}{y} \end{aligned}

Hence order of differential equation will be ‘2’ i.e. 2nd order.

So, the answer is ‘2’ or 2nd order.