Question

For each of the following differential equations find the particular solution.

(x − y² x) dx − (y + x² y) dy = 0, when x = 2, y = 0

Answer 1

(x − y² x)dx − (y + x² y) dy = 0, when x = 2, y = 0

∴ x(1- y²) dx = y(1 + x² ) dy

∴ `(xdx)/(1+x^2) = (ydy)/(1-y^2)`

Integrating on both sides, we get

`int( 2x)/(1+x^2) dx = int(2y)/(1-y^2 )dy`

∴ `int( 2x)/(1+x^2) dx = - int(-2y)/(1-y^2 )dy`

∴ `log | 1 + x^2| = -log| 1-y^2| + log |c|`

∴ `log |1 + x^2 | = log |c /(1-y^2)|`

∴  (1 + x ²) ( 1 - y² ) = c  …(i)

When x = 2, y = 0, we have

(1 + 4) (1 - 0) = c

∴  c = 5

Substituting c = 5 in (i),we get

(1 + x²) ( 1-y² ) = 5,

which is the required particular solution.