If $y = 3e^{2x} + 2e^{3x}$, then $\frac{d^2y}{dx^2} + 6y$ is equal to

Given $y = 3e^{2x} + 2e^{3x}$, find $\frac{d^2y}{dx^2} + 6y$.

When differentiating $e^{ax}$, the derivative is $a \cdot e^{ax}$.

$\frac{dy}{dx} = 3 \cdot (2e^{2x}) + 2 \cdot (3e^{3x})$

$\frac{dy}{dx} = 6e^{2x} + 6e^{3x}$

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Find the second derivative:

$\frac{d^2y}{dx^2} = 6 \cdot (2e^{2x}) + 6 \cdot (3e^{3x})$

$\frac{d^2y}{dx^2} = 12e^{2x} + 18e^{3x}$

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Calculate $\frac{d^2y}{dx^2} + 6y$:

$\frac{d^2y}{dx^2} + 6y = (12e^{2x} + 18e^{3x}) + 6(3e^{2x} + 2e^{3x})$

$= 12e^{2x} + 18e^{3x} + 18e^{2x} + 12e^{3x}$

$= 30e^{2x} + 30e^{3x}$

$= 30(e^{2x} + e^{3x})$

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From the first derivative, $\frac{dy}{dx} = 6e^{2x} + 6e^{3x} = 6(e^{2x} + e^{3x})$

Therefore:

$30(e^{2x} + e^{3x}) = 5 \times 6(e^{2x} + e^{3x})$

$= 5\frac{dy}{dx}$

Therefore, $\frac{d^2y}{dx^2} + 6y = 5\frac{dy}{dx}$