The differential equation of $y=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x)$ is

The differential equation of $y=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x)$ is
  1. $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}-y=0$
  2. $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}+2 \frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=0$
  3. $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+y=0$
  4. $\frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=0$

Solution

$\begin{aligned} & y=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x) \\ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\mathrm{e}^x(\mathrm{a} \cos x+\mathrm{b} \sin x) \\ & +\mathrm{e}^x(\mathrm{~b} \cos x-\mathrm{a} \sin x) \\ & \Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=y+\mathrm{e}^x(\mathrm{~b} \cos x-\mathrm{a} \sin x) \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=\frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{e}^x(\mathrm{~b} \cos x-\mathrm{a} \sin x) \\ & +\mathrm{e}^x(-\mathrm{b} \sin x-\mathrm{a} \cos x) \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}=\frac{\mathrm{d} y}{\mathrm{~d} x}+\left(\frac{\mathrm{d} y}{\mathrm{~d} x}-y\right)-y \\ & \Rightarrow \frac{\mathrm{d}^2 y}{\mathrm{~d} x^2}-2 \frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=0 \\ & \end{aligned}$

Asked in: MHT CET 2023 (14 May Shift 1)

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