Question

Solve the following differential equation:

`"dy"/"dx" = "e"^("x + y") + "x"^2 "e"^"y"`

Answer 1

∴ `"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`

∴ 3e^x + 3e^-y + x³ = - 3c₁

∴ 3e^x + 3e^-y + x³ = c, where c =  - 3c₁ 

This is the general solution.