The differential equation of the family $y=a e^x+b x e^x+c x^2 e^x$ of curves, where $a, b, c$ are arbitrary constants, is

• A

  $y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0$
• B

  $y^{\prime \prime \prime}+3 y^{\prime \prime}-3 y^{\prime}-y=0$
• C

  $y^{\prime \prime \prime}-3 y^{\prime \prime}-3 y^{\prime}+y=0$
• Correct answer
  D

  $y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$

Solution

On differentiating w.r.t. $x$, we get $\begin{aligned} y^{\prime} & =a e^x+b\left(x e^x+e^x\right)+c\left(x^2 e^x+2 x e^x\right) \\ \Rightarrow \quad y^{\prime} & =a e^x+b x e^x+c x^2 e^x+b e^x+2 c x e^x \end{aligned}$
Again differentiating w.r.t. $x$, we get $\begin{aligned} & y^{\prime \prime}=y^{\prime}+b e^x+2 c\left(x e^x+e^x\right) \\ \Rightarrow \quad y^{\prime \prime} & =y^{\prime}+b e^x+2 c x e^x+2 c e^x \end{aligned}$
$\Rightarrow \quad y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0$

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