Question

The solution of the differential equation y₁ y₃ = y₂² is
(a) x = C₁ e^C₂y + C₃
(b) y = C₁ e^C₂x + C₃
(c) 2x = C₁ e^C₂y + C₃
(d) none of these

Answer 1

(b) y = C₁ e^C₂x + C₃

$$\begin{gathered} y_1{y}_3=y_2^2 \\ \frac{y_3}{y_2}=\frac{y_2}{y_1} \\ \Rightarrow \frac{\left({\displaystyle \frac{d^{3}y}{d{x}^3}}\right)}{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}=\frac{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}{\left({\displaystyle \frac{dy}{dx}}\right)} \\ \Rightarrow \int \frac{{\displaystyle \frac{d}{dx}\left(\frac{d^{2}y}{d{x}^2}\right)}}{\left({\displaystyle \frac{d^{2}y}{d{x}^2}}\right)}=\int \frac{{\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)}}{\left({\displaystyle \frac{dy}{dx}}\right)} \\ \Rightarrow \ln \left(\frac{d^{2}y}{d{x}^2}\right)=\ln \left(\frac{dy}{dx}\right)+\ln C_4 \\ \Rightarrow \frac{d^{2}y}{d{x}^2}=C_4\frac{dy}{dx} \\ \Rightarrow \int \frac{{\displaystyle \frac{d}{dx}\left(\frac{dy}{dx}\right)}}{\left({\displaystyle \frac{dy}{dx}}\right)}=\int C_{4}dx \\ \ln \left(\frac{dy}{dx}\right)=C_{4}x+C_5 \\ \Rightarrow \frac{dy}{dx}=e^{C_{4}x+C_5} \\ \int dy=\int \left(e^{C_{4}x+C_5}\right)dx \\ y=\frac{e^{C_{4}x+C_5}}{C_4}+C_6 \\ y=\frac{e^{C_{4}x}.e^{C_5}}{C_4}+C_6 \\ \Rightarrow y=C_1{e}^{C_{2}x}+C_3 \\ \mathrm{where}, \\ {C}_1=\frac{e^{C_5}}{C_4} \\ {C}_4=C_2 \\ {C}_6=C_3 \end{gathered}$$