Question

If y = 3 e^2x + 2 e^3x, prove that  \[\frac{d^2 y}{d x^2} - 5\frac{dy}{dx} + 6y = 0\] ?

Answer 1

Here,

\[y = 3 e^{2x} + 2 e^{3x} \]

\[\text {Differentiating w . r . t . x, we get }\]

\[\frac{d y}{d x} = 6 e^{2x} + 6 e^{3x} \]

\[\text { Differentiating again w . r . t . x, we get }\]

\[\frac{d^2 y}{d x^2} = 12 e^{2x} + 18 e^{3x} \]

\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( 6 e^{2x} + 6 e^{3x} \right) - 6\left( 3 e^{2x} + 2 e^{3x} \right)\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = 5\left( \frac{dy}{dx} \right) - 6y\]

\[ \Rightarrow \frac{d^2 y}{d x^2} - 5\left( \frac{dy}{dx} \right) + 6y = 0\]