If $y=3 e^{5 x}+5 e^{3 x}, \quad$ then $\frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}=$
If $y=3 e^{5 x}+5 e^{3 x}, \quad$ then $\frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x}=$
- $-10 y$
- $15 y$
- $-15 y$
- $10 y$
Solution
$\begin{aligned} & y=3 e^{5 x}+5 e^{3 x} \\ \therefore & \frac{d y}{d x}=3 e^{5 x} \times 5+5 e^{3 x} \times 3=15 e^{5 x}+15 e^{3 x} \\ \frac{d^{2} y}{d x^{2}} &=15 e^{5 x} \times 5+15 e^{3 x} \times 3 \\ &=75 e^{5 x}+45 e^{3 x} \\ \therefore \frac{d^{2} y}{d x^{2}}-8 \frac{d y}{d x} &=75 e^{5 x}+45 e^{3 x}-8\left(15 e^{5 x}+15 e^{3 x}\right) \\ &=75 e^{5 x}+45 e^{3 x}-120 e^{5 x}-120 e^{3 x} \\ &=-45 e^{5 x}-75 e^{3 x} \\ &=-15\left(3 e^{5 x}+5 e^{3 x}\right) \\ &=-15 y \end{aligned}$
Asked in: MHT CET 2020 (12 Oct Shift 1)
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