Discussion Forum

1)

 If curve y=f(x) passes through the point (1,-1) and satisfies the differential equation

y(1+xy)dx=xdy , then f(-1/2) is equal to 

Answer:

Option D

Explanation:

Given  differential equation is 

y(1+xy)dx=xdy

$\Rightarrow y dx+xy^{2}dx=x dy$

$\Rightarrow \frac{x dy -y dx}{y^{2}}=xdx$

$\Rightarrow -\frac{(y dx -x dy)}{y^{2}}=xdx$

$\Rightarrow -d(\frac{x}{y})=xdx$

 On  integrating  both sides , we get

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

 It passes through(1,-1)

$1=\frac{1}{2}+C$

$\Rightarrow C=\frac{1}{2}$

Now, from (i)

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

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

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

$f(-\frac{1}{2})=\frac{4}{5}$