Let y - yx be the solution of the differential equation ex-1-y2 dx-yxdy-0- y1-1- Then the value of y32 is equal to

E^x √(1 y^2)dx= yxdy ⇒∫ xe^xdx=∫ y√(1 y^2)dy ⇒ xe^x e^x=√(1 y^2)+c y(1)= 1 ⇒ 0=0+c⇒ c=0 ∴ xe^x e^x=√(1 y^2) for y(3) put x=3 3e^3 e^3=√(1 y^2) ⇒ 4e^6=1 y^2 ⇒ (y ...

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Byju's Answer

Standard XII

Mathematics

Integrating Factor

Let y = yx be...

Question

Let $y = y (x)$ be the solution of the differential equation $e^{x} \sqrt{1 - y^{2}} d x + (\frac{y}{x}) d y = 0,$ $y (1) = - 1.$ Then the value of $(y (3))^{2}$ is equal to

1 + 4 e^{6}

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1 - 4 e^{6}

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1 - 4 e^{3}

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1 + 4 e^{3}

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Solution

The correct option is B $1 - 4 e^{6}$
$e^{x} \sqrt{1 - y^{2}} d x = - \frac{y}{x} d y$
$\Rightarrow \int x e^{x} d x = \int \frac{- y}{\sqrt{1 - y^{2}}} d y$
$\Rightarrow x e^{x} - e^{x} = \sqrt{1 - y^{2}} + c$
$y (1) = - 1$
$\Rightarrow 0 = 0 + c \Rightarrow c = 0$
$∴ x e^{x} - e^{x} = \sqrt{1 - y^{2}}$
for $y (3)$ put $x = 3$
$3 e^{3} - e^{3} = \sqrt{1 - y^{2}}$
$\Rightarrow 4 e^{6} = 1 - y^{2}$
$\Rightarrow (y (3))^{2} = 1 - 4 e^{6}$

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Similar questions

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Let y=y(x) be the solution of the differential equation ex√1−y2 dx+(yx)dy=0, y(1)=−1. Then the value of (y(3))2 is equal to

The correct option is B 1−4e6 ex√1−y2dx=−yxdy ⇒∫xexdx=∫−y√1−y2dy ⇒xex−ex=√1−y2+c y(1)=−1 ⇒0=0+c⇒c=0 ∴xex−ex=√1−y2 for y(3) put x=3 3e3−e3=√1−y2 ⇒4e6=1−y2 ⇒(y(3))2=1−4e6

Let $y = y (x)$ be the solution of the differential equation $e^{x} \sqrt{1 - y^{2}} d x + (\frac{y}{x}) d y = 0,$ $y (1) = - 1.$ Then the value of $(y (3))^{2}$ is equal to