Question

\[\frac{dy}{dx} + 2y = 6 e^x\]

Answer 1

We have, 
\[\frac{dy}{dx} + 2y = 6 e^x . . . . . \left( 1 \right)\]
Clearly, it is a linear differential equation of the form 
\[\frac{dy}{dx} + Py = Q\]
where
\[P = 2\]
\[Q = 6 e^x \]
\[ \therefore \text{I.F.} = e^{\int P dx} \]
\[ = e^{\int2 dx} \]
\[ = e^{2x} \]
\[\text{ Multiplying both sides of }\left( 1 \right)\text{ by }e^{2x} ,\text{ we get }\]
\[ e^{2x} \left( \frac{dy}{dx} + 2y \right) = 6 e^{2x} e^x \]
\[ \Rightarrow e^{2x} \frac{dy}{dx} + 2 e^{2x} y = 6 e^{3x} \]
\[\text{ Integrating both sides with respect to x, we get }\]
\[y e^{2x} = 6\int e^{3x} dx + C\]
\[ \Rightarrow y e^{2x} = 6\frac{e^{3x}}{3} + C\]
\[ \Rightarrow y e^{2x} = 2 e^{3x} + C\]
\[\text{ Hence, }y e^{2x} = 2 e^{3x} + C\text{ is the required solution . }\]