Q118 / 160 · TS EAMCET 2020 (09 Sep Shift 2) · Mathematics

The differential equation of which $x y=a e^x+b e^{-x}+x^2$ is a solution, is

  • A
    $x y^{\prime \prime}-2 y^{\prime}+x y+x^2-2=0$
  • B
    $x y^{\prime \prime}+2 y^{\prime}-x+x^2+2=0$
  • C
    $x y^{\prime \prime}+2 y^{\prime}-y+x^2-2=0$
  • Correct answer
    D
    $x y^{\prime \prime}+2 y^{\prime}-x y+x^2-2=0$

Solution

We have, $x y=a e^x+b e^{-x}+x^2$ $\therefore \quad y+x \frac{d y}{d x}=a e^x-b e^{-x}+2 x$ $\Rightarrow \quad \frac{d y}{d x}+x \frac{d^2 y}{d x^2}+\frac{d y}{d x}=a e^x+b e^{-x}+2$ From Eqs. (i)and (iii), we get $x \frac{d^2 y}{d x^2}+2 \frac{d y}{d x}=x y-x^2+2$ $\Rightarrow \quad x y^{\prime \prime}+2 y^{\prime}-x y+x^2-2=0$

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

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