If y = 3e^ 2x + 2e^ 3x , prove that d^ 2 y dx^ 2 -5 dy dx +6y=0. - Vidyadip

Question

If $y = 3e^{2x }+ 2e^{3x},$ prove that $\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-5\frac{\text{dy}}{\text{dx}}+\text{6y}=0.$

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Answer

$\frac{\text{dy}}{\text{dx}}=6\text{ e}^{\text{2x}}+6\cdot\text{e}^{\text{3x}}$
$\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}=12\text{ e}^{\text{2x}}+18\cdot\text{e}^{\text{3x}}$
$\Rightarrow\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}}-5\frac{\text{dy}}{\text{dx}}+6\text{y}$
$= (12 e^{2x }+ 18 e ^{3x}) -5 (6 e^{2x}+ 6 e^{3x }) + 6 (3 e^{2x }+ 2 e^{3x})$
$= 30 e^{2x} - 30 e^{2x} + 30 e^{3x} - 30 e^{3x} $
$= 0.$

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