Question
Question: If y¢ = \(\frac{x - y}{x + y}\), then its solution is –...
If y¢ = x+yx−y, then its solution is –
A
y2 + 2xy – x2 = c
B
y2 + 2xy + x2 = c
C
y2 – 2xy – x2 = c
D
y2 – 2xy + x2 = c
Answer
y2 + 2xy – x2 = c
Explanation
Solution
Given, dxdy=x+yx−y This is a homogeneous equation
Put y = vx Ž dxdy = v + x dxdv
Given equation becomes
v + x dxdv = 1+v1−v
Ž xdxdv = 1+v1−v – v
Ž 2−(1+v)21+v dv = xdx
On integrating both sides
∫2−(1+v)21+v dv = ∫xdx
Put (1 + v)2 = t Ž 2(1 + v) dv = dt
Ž 21 ∫2−tdt = ∫xdx
Ž –21log (2 – t) = log x + log c
Ž –21log [2 – (1 + v)2] = log xc
Ž –21 log [–v2 – 2v + 1] = log xc
Ž log 1−2v−v21 = log xc
Ž x2c2 (1 – 2v – v2) = 1
Ž x2c2 (1−x2y−x2y2) = 1[∵v=xy]
Ž x2x2c2(x2−2yx−y2) = 1
Ž y2 + 2xy – x2 = c