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Question
If y = 3 e2x + 2 e3x, prove that \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?
Solution
Here,
\[y = 3 e^{2x} + 2 e^{3x} \]
\[\text {Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = 6 e^{2x} + 6 e^{3x} \]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 12 e^{2x} + 18 e^{3x} \]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( 6 e^{2x} + 6 e^{3x} \right) - 6\left( 3 e^{2x} + 2 e^{3x} \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( \frac{dy}{dx} \right) - 6y\]
\[ \Rightarrow \frac{d^2 y}{d x^2} - 5\left( \frac{dy}{dx} \right) + 6y = 0\]
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