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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
MCQ
Choose the correct answer from the given four options.The solution of the differential equation $\frac{\text{dy}}{\text{dx}}+\frac{2\text{xy}}{1+\text{x}^2}=\frac{1}{(1+\text{x}^2)^2}$ is:
  • ✓
    $\text{y}(1+\text{x}^2)=\text{C}+\tan^{-1}\text{x}$
  • B
    $\frac{\text{y}}{1+\text{x}^2}=\text{C}+\tan^{-1}\text{x}$
  • C
    $\text{y}\log(1+\text{x}^2)=\text{C}+\tan^{-1}\text{x}$
  • D
    $\text{y}(1+\text{x}^2)=\text{C}+\sin^{-1}\text{x}$
✓

Answer

Correct option: A.
$\text{y}(1+\text{x}^2)=\text{C}+\tan^{-1}\text{x}$
Given is, $\frac{\text{dy}}{\text{dx}}+\frac{2\text{xy}}{1+\text{x}^2}=\frac{1}{(1+\text{x}^2)^2}$
Here, $\text{P}=\frac{2\text{x}}{1+\text{x}^2}$ and $\text{Q}=\frac{1}{(1+\text{x}^2)^2}$
This ia a linerar differential equation.
$\therefore\text{I.F.}=\text{e}^{\int\frac{2\text{x}}{1+\text{x}^2}\text{dx}}$
Put $1+\text{x}^2=\text{t}$
$\Rightarrow2\text{x}\text{ dx}=\text{dt}$
$\therefore\text{I.F.}=\text{e}^{\int\frac{\text{dt}}{\text{t}}}=\text{e}^{\log\text{t}}$
$=\text{e}^{\log(1+\text{x}^2)}=1+\text{x}^2$
Thus, the general solution is
$\text{y}.(1+\text{x}^2)=\int(1+\text{x}^2)\frac{1}{(1+\text{x}^2)}+\text{C}$
$\Rightarrow\text{y}(1+\text{x}^2)=\int\frac{1}{(1+\text{x}^2)}\text{dx}+\text{C}$
$\Rightarrow\text{y}(1+\text{x}^2)=\tan^{-1}\text{x}+\text{C}$

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