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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
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Solution
(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.
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