Question

The solution of the differential equation y₁ y₃ = y₂² is

Options

• x = C₁ e^C₂y + C₃

• y = C₁ e^C₂x + C₃

• 2x = C₁ e^C₂y + C₃

• none of these

Answer 1

y = C₁ e^C₂x + C₃

 

\[y_1 y_3 = y_2^2 \]
\[\frac{y_3}{y_2} = \frac{y_2}{y_1}\]
\[ \Rightarrow \frac{\left( \frac{d^3 y}{d x^3} \right)}{\left( \frac{d^2 y}{d x^2} \right)} = \frac{\left( \frac{d^2 y}{d x^2} \right)}{\left( \frac{dy}{dx} \right)}\]
\[ \Rightarrow \int\frac{\frac{d}{dx}\left( \frac{d^2 y}{d x^2} \right)}{\left( \frac{d^2 y}{d x^2} \right)} = \int\frac{\frac{d}{dx}\left( \frac{dy}{dx} \right)}{\left( \frac{dy}{dx} \right)}\]
\[ \Rightarrow \ln\left( \frac{d^2 y}{d x^2} \right) = \ln\left( \frac{dy}{dx} \right) + \ln C_4 \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = C_4 \frac{dy}{dx}\]
\[ \Rightarrow \int\frac{\frac{d}{dx}\left( \frac{dy}{dx} \right)}{\left( \frac{dy}{dx} \right)} = \int C_4 dx\]
\[\ln\left( \frac{dy}{dx} \right) = C_4 x + C_5 \]
\[ \Rightarrow \frac{dy}{dx} = e^{C_4 x + C_5} \]
\[\int dy = \int \left( e^{C_4 x + C_5} \right) dx\]
\[y = \frac{e^{C_4 x + C_5}}{C_4} + C_6 \]
\[y = \frac{e^{C_4 x} . e^{C_5}}{C_4} + C_6 \]
\[ \Rightarrow y = C_1 e^{C_2 x} + C_3 \]
where, 
\[ C_1 = \frac{e^{C_5}}{C_4}\]
\[ C_4 = C_2 \]
\[ C_6 = C_3 \]