Question

Form the differential equation representing the family of curves y = A sin x, by eliminating the arbitrary constant A.

Answer 1

Given: y = A sinx


$$\begin{gathered} y=A\sin x...\left(1\right) \\ \mathrm{Differentiating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x \\ \Rightarrow \frac{dy}{dx}=A\cos x \\ \Rightarrow A=\frac{1}{\cos x}\frac{dy}{dx} \\ \mathrm{Substituting}\mathrm{the}\mathrm{value}\mathrm{of}A\mathrm{in}\left(1\right),\mathrm{we}\mathrm{get} \\ y=\frac{1}{\cos x}\frac{dy}{dx}\sin x \\ \Rightarrow y=\frac{\sin x}{\cos x}\frac{dy}{dx} \\ \Rightarrow y=\tan x\frac{dy}{dx} \\ \Rightarrow \tan x\frac{dy}{dx}-y=0 \end{gathered}$$

Hence, the differential equation representing the family of curves y = A sinx is $\tan x\frac{dy}{dx}-y=0.$