Q119 / 160 · AP EAMCET 2022 (07 Jul Shift 2) · Mathematics

An integrating factor of the differential equation $\left(x^2+1\right) \frac{d y}{d x}+x y=x^3$ is

  • A
    $\frac{x}{1+x^2}$
  • B
    $\frac{1}{2} \log \left(1+x^2\right)$
  • Correct answer
    C
    $\sqrt{1+x^2}$
  • D
    $e^{\log \left(1+x^2\right)}$

Solution

Differential equation is $$ \begin{aligned} & \frac{d y}{d x}+\left(\frac{x}{1+x^2}\right) y=\frac{x^3}{x^2+1} \\ & \mathrm{IF}=e^{\left(\int \frac{x}{1+x^2} d x\right)}=e^{\left(\frac{1}{2} \ln \left|1+x^2\right|\right)} \\ & =e^{\ln \left(\sqrt{x^2+1}\right.}=\sqrt{x^2+1} \end{aligned} $$

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

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