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CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY3 Marks
Question
If $\text{y}=3\text{e}^{2\text{x}}+2\text{e}^{3\text{x}}$ prove that $\frac{\text{d}^2\text{y}}{\text{dx}^2}-5\frac{\text{dy}}{\text{dx}}+6\text{y}=0$
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Answer

$\text{y}=3\text{e}^{2\text{x}}+2\text{e}^{3\text{x}}$
Differentiating w.r.t.x, we get
$\frac{\text{dy}}{\text{dx}}=6\text{e}^{2\text{x}}+6\text{e}^{3\text{x}}$
Differentiating w.r.t.x, we get
$\frac{\text{d}^2\text{y}}{\text{dx}^2}=12\text{e}^{2\text{x}}+18\text{e}^{3\text{x}}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=5(6\text{e}^{2\text{x}}+6\text{e}^{3\text{x}})-6(3\text{e}^{2\text{x}}+2\text{e}^{3\text{x}})$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}=5\Big(\frac{\text{dy}}{\text{dx}}\Big)-6\text{y}$
$\Rightarrow\frac{\text{d}^2\text{y}}{\text{dx}^2}-5\Big(\frac{\text{dy}}{\text{dx}}\Big)+6\text{y}=0$

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