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प्रश्न
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is ______.
पर्याय
y = tan–1x
y – x = k(1 + xy)
x = tan–1y
tan(xy) = k
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उत्तर
The solution of the differential equation `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)` is y – x = k(1 + xy).
Explanation:
The given differential equation is `("d"y)/("d"x) + (1 + y^2)/(1 + x^2)`
⇒ `("d"y)/(1 + y^2) = ("d"x)/(1 + x^2)`
Integrating both sides, we get
`int ("d"y)/(1 + y^2) = int ("d"x)/(1 + x^2)`
⇒ tan–1y = tan–1x + c
⇒ tan–1y – tan–1x = c
⇒ `tan^-1((y - x)/(1 + xy))` = c
⇒ `(y - x)/(1 + xy)` = tan c
⇒ `((y - x)/(1 + xy))` = k ....[k = tan c]
⇒ y – x = k(1 + xy)
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