JEE MAIN 2016

Calculus Level 4

If a curve y = f ( x ) y=f(x) passes through the point ( 1 , − 1 ) (1,-1) and satisfies the differential equation y ( 1 + x y ) d x = x d y y(1+xy) \, dx= x \, dy . What is the value of f ( − 1 2 ) f\left(-\dfrac{1}{2}\right) ?

2 5 \frac 25 4 5 \frac 45 − 2 5 -\frac 25 − 4 5 -\frac 45

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Md Zuhair by Md Zuhair , 20, India

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2 solutions

Kushal Bose
Jul 25, 2016

From the given differential equation :

y d x + x y 2 d x = x d y ydx+xy^2dx = xdy

y d x − x d y = − x y 2 d x ydx-xdy = -xy^2dx

y d x − x d y y 2 = − x d x \dfrac{ydx-xdy}{y^2} = -xdx

d ( x y ) = − x d x d(\dfrac{x}{y}) = -xdx

∫ d ( x y ) = − ∫ x d x \int d(\dfrac{x}{y})= -\int xdx

x y = − x 2 2 + c \dfrac{x}{y}= -\dfrac{x^2}{2} + c where c is an integration constant.

From the given info now we can get the value c and also find the f(-1/2)

3 Helpful 0 Interesting 0 Brilliant 0 Confused

First of all one has to realize that a Bernoulli differential equation with the knwn entry z=1/y has to be solved!.

Andreas Wendler - 4 years, 10 months ago
Md Zuhair
Jul 24, 2016

now We will be finding the equation f(x) by solving the differential eq. bold y(1+xy).dx= x.dy .. Hence now Solving it and putting (1,-1) here we get f(x)= \frac{-2x}{1+x^{2}} .. Now put the value of f(-1/2) to get your ans,

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