Solve the differential equation dydx=ex+y+x2ey. - Mathematics and Statistics

प्रश्न

Solve the differential equation \[\frac{dy}{dx} = e^{x + y} + x^2 e^y\].

बेरीज

उत्तर

`("d"y)/("d"x)` = e^(x+y) + x²e^y

∴ `("d"y)/("d"x)` = e^x . e^y + x² e^y

∴ `("d"y)/("d"x)` = e^y(e^x + x²)

∴ `("d"y)/("e"^y)` = (e^x + x²) dx

Integrating on both sides, we get

`int_"e"^(-y) "d"y = int("e"^x + x^2) "d"x`

∴ `("e"^(-y))/(-1) = "e"^x + x^3/3 + "c"_1`

∴ e^−y = `- "e"^x - x^3/3 - "c"_1`

∴ `"e"^(-y) + "e"^x + x^3/3` = c, where c = c₁

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