1.

Solve the differential equation dy/dx = (x2 + y2)/xy given that y(1) = 2.

Answer»

dy/dx = \(\frac{(x^2+y^2)}{xy}\); y(1) = 2

dy/dx = \(\frac{1+(y/x)^2}{y/x}\)  _________(1)

Let y = vx.

Then dy/dx = v + x dv/dx

Then from (1), we get

v + x dv/dx = \(\frac{1+v^2}v\)

⇒ x dv/dx = \(\frac{1+v^2}v\) - v = \(\frac{1+v^2-v^2}v\) = 1/v

⇒ v dv = 1/x dx

⇒ ∫ v dv = ∫ 1/x dx

⇒ v2/2 = log x + log c (where log c is an integral constant)

⇒ v2 = 2 log cx

⇒ (y/x)2 = 2 log cx

⇒ y2 = 2 x2 log cx   _________(2)

∵ y(1) = 2

∴ 4 = 2 log c

⇒ log c = 4/2 = 2

⇒ c = c2

∴ From (2), y2 = 2 x2 log (e2x), which represent the solution of given differential equation.


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