Question

The tangent at any point (x, y) of a curve makes an angle tan⁻¹(2x + 3y) with x-axis. Find the equation of the curve if it passes through (1, 2).

Answer 1

The slope of the curve is given as $\frac{dy}{dx}=\tan \theta$.
Here,

$$\begin{gathered} \theta ={\tan }^{-1}\left(2x+3y\right) \\ \therefore \frac{dy}{dx}=\tan \left({\tan }^{-1}2x+3y\right) \\ \Rightarrow \frac{dy}{dx}=2x+3y \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dy}{dx}-3y=2x.....\left(1\right) \\ \mathrm{Clearly},\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{linear}\mathrm{differential}\mathrm{equation}\mathrm{of}\mathrm{the}\mathrm{form} \\ \frac{dy}{dx}+Py=Q \\ \mathrm{where}P=-3\mathrm{and}Q=2x \\ \therefore \mathrm{I}.\mathrm{F}.=e^{\int Pdx} \\ =e^{\int -3dx} \\ =e^{-3x} \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{of}\left(1\right),\mathrm{by}\mathrm{I}.\mathrm{F}.=e^{-3x},\mathrm{we}\mathrm{get} \\ {e}^{-3x}\left(\frac{dy}{dx}-3y\right)=e^{-3x}.2x \\ \Rightarrow e^{-3x}\left(\frac{dy}{dx}-3y\right)=2x{e}^{-3x} \\ \mathrm{Integrating}\mathrm{both}\mathrm{sides}\mathrm{with}\mathrm{respect}\mathrm{to}x,\mathrm{we}\mathrm{get} \\ y{e}^{-3x}=2\int x{e}^{-3x}dx+C \\ \Rightarrow y{e}^{-3x}=2\int \underset{\mathrm{I}}{x}\underset{\mathrm{II}}{e^{-3x}}dx+C \\ \Rightarrow y{e}^{-3x}=2x\int e^{-3x}dx-2\int \left[\frac{d}{dx}\left(x\right)\int e^{-3x}dx\right]dx+C \\ \Rightarrow y{e}^{-3x}=-2x\frac{e^{-3x}}{3}+2\times \frac{1}{3}\int e^{-3x}dx+C \\ \Rightarrow y{e}^{-3x}=-\frac{2}{3}x{e}^{-3x}-2\times \frac{1}{9}{e}^{-3x}+C \\ \Rightarrow y{e}^{-3x}=-\frac{2}{3}x{e}^{-3x}-\frac{2}{9}{e}^{-3x}+C \\ \mathrm{Since}\mathrm{the}\mathrm{curve}\mathrm{passes}\mathrm{through}\left(1,2\right),\mathrm{it}\mathrm{satisfies}\mathrm{the}\mathrm{above}\mathrm{equation}. \\ \therefore 2e^{-3}=-\frac{2}{3}{e}^{-3}-\frac{2}{9}{e}^{-3}+C \\ \Rightarrow C=2e^{-3}+\frac{2}{3}{e}^{-3}+\frac{2}{9}{e}^{-3} \\ \Rightarrow C=\frac{26}{9}{e}^{-3} \\ \mathrm{Putting}\mathrm{the}\mathrm{value}\mathrm{of}C,\mathrm{we}\mathrm{get} \\ y{e}^{-3x}=\left(-\frac{2}{3}x-\frac{2}{9}\right)e^{-3x}+\frac{26}{9}{e}^{-3} \end{gathered}$$