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if a curve y=f(x) passes through the point (1,-1)satisfies the differential equation y(1+xy) dx= x dy then f(-1/2)

if a curve y=f(x) passes through the point (1,-1)satisfies the differential equation y(1+xy) dx= x dy then f(-1/2)

Grade:12

2 Answers

Aditya Gupta
2081 Points
3 years ago
write it as
ydx + xy^2dx= xdy
or (ydx – xdy)/y^2 = – xdx
now note that (ydx – xdy)/y^2 = d(x/y)
so d(x/y)= – xdx
integrate both sides
x/y= – x^2/2 + C
(1,-1)satisfies the differential equation so
 – 1= – ½ + C
or C= – ½ 
so, y=f(x)= – 2x/(1+x^2)
so f(-1/2)= 4/5= 0.8
kindly approve :))
Vikas TU
14149 Points
3 years ago
Dear student 
y/x (1+xy) = dy/dx 
y = vx = y/x = v 
dy/dx = v+xdv/dx 
v(1+vx^2) = v+xdv/dx 
v^2x^2 = xdv/dx 
Integrate this 
x^2/2 = -1/v + c 
x^2/2 = -x/y - 1/2 
f(-1/2) , put x = -1/2 
1/8 = 1/2y - 1/2 
y = 4/5 

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