Q119 / 160 · AP EAMCET 2023 (16 May Shift 2) · Mathematics

The differential equation having $y=(a+b) e^{c x+d}$ as its general solution, where a, b, c, d, are arbitrary constants, is

  • A
    $y^{(4)}+3 y y^{(3)}+6 y^{(2)} y^2+y=0$
  • B
    $\mathrm{y}^{(3)}+4 \mathrm{yy} \mathrm{y}^{(2)}+6 \mathrm{y}^2 \mathrm{y}^{(1)}+12 \mathrm{y}=0$
  • C
    $\mathrm{y}^{(1)}-\mathrm{y}=0$
  • Correct answer
    D
    $\mathrm{yy}^{(2)}-\left(\mathrm{y}^{(1)}\right)^2=0$

Solution

Given $y=(a+b) e^{c x+d}$ $y^1=(a+b) e^{(x+d)}$ $\begin{aligned} & y^1=c y ; \quad y^2=c y^1 \\ & y^2=\frac{y^{\prime}}{y} \cdot y^1 \\ & \Rightarrow y y^{(2)}-\left(y^{(1)}\right)^2=0\end{aligned}$

Hence, the correct answer is option: 'D'.

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