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Question
Solve the following differential equation:
`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`
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Solution
∴ `"dy"/"dx" = "e"^"x" * "e"^"y" + "x"^2 * "e"^"y" = "e"^"y"("e"^"x" + "x"^2)`
∴ `1/"e"^"y" "dy" = ("e"^"x" + "x"^2)`dx
Integrating both sides, we get
`int "e"^(- "y") "dy" = int("e"^"x" + "x"^2)`dx
∴ `"e"^(-"y")/-1 = "e"^"x" + "x"^3/3 + "c"_1`
∴ `"e"^"x" + "e"^(-"y") + "x"^3/3 = - "c"_1`
∴ 3ex + 3e-y + x3 = - 3c1
∴ 3ex + 3e-y + x3 = c, where c = - 3c1
This is the general solution.
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