The general solution of the differential equation $\sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \frac{d y}{d x}=0$ is: (where C is a constant of integration)

(A) $\sqrt{1+\mathrm{y}^{2}}+\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}+1}{\sqrt{1+\mathrm{x}^{2}}-1}\right)+\mathrm{C}$

(B) $\sqrt{1+\mathrm{y}^{2}}-\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}+1}{\sqrt{1+\mathrm{x}^{2}}-1}\right)+\mathrm{C}$

(C) $\sqrt{1+\mathrm{y}^{2}}-\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}+1}{\sqrt{1+\mathrm{x}^{2}}-1}\right)+\mathrm{C}$

(D) $\sqrt{1+\mathrm{y}^{2}}+\sqrt{1+\mathrm{x}^{2}}=\frac{1}{2} \log _{\mathrm{e}}\left(\frac{\sqrt{1+\mathrm{x}^{2}}-1}{\sqrt{1+\mathrm{x}^{2}}+1}\right)+\mathrm{C}$

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Differential Equations

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The general solution of the differential equation

1 + x^{2} + y^{2} + x^{2} y^{2} + x y \frac{d y}{d x} = 0

is: (where C is a constant of integration)

(A) $1 + y^{2} + 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$ ✓ Correct
(B) $1 + y^{2} - 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$
(C) $1 + y^{2} - 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} + 1}{1 + x ^{2} - 1}) + C$
(D) $1 + y^{2} + 1 + x^{2} = \frac{1}{2} lo g_{e} (\frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}) + C$

Answer: (A)

Solution

Group the terms under the root:

1 + x^{2} + y^{2} + x^{2} y^{2} = (1 + x^{2}) (1 + y^{2})

, so the equation is

(1 + x^{2}) (1 + y^{2}) + x y \frac{d y}{d x} = 0

and the variables separate as

\frac{y}{1 + y ^{2}} d y = - \frac{1 + x ^{2}}{x} d x

. The left side integrates to

1 + y^{2}

, while

\int \frac{1 + x ^{2}}{x} d x = 1 + x^{2} + \frac{1}{2} ln \frac{1 + x ^{2} - 1}{1 + x ^{2} + 1}

. Carrying the minus sign across and collecting the logarithm, the general solution is

1 + y^{2} + 1 + x^{2} = \frac{1}{2} ln \frac{1 + x ^{2} + 1}{1 + x ^{2} - 1} + C

JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Single correct option

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JEE Main · 2020 · 06 Sep 2020 · Shift 1 · Differential Equations

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