# Solve the following differential equation: y dx + (x - y2) dy = 0 - Mathematics and Statistics

Sum

Solve the following differential equation:

y dx + (x - y2) dy = 0

#### Solution

y dx + (x - y2) dy = 0

∴ y dx = - (x - y2) dy

∴ "dx"/"dy" = - (("x - y"^2))/"y" = - "x"/"y" + "y"

∴ "dx"/"dy" + (1/"y") * "x" = "y"     ....(1)

This is the linear differential equation of the form

"dx"/"dy" + "P" * "x" = "Q", where P = 1/"y" and Q = y

∴ I.F. = "e"^(int "P dy") = "e"^(int 1/"y" "dy") = "e"^(log "y") = "y"

∴ the solution of (1) is given by

"x" * ("I.F.") = int "Q" * (I.F.) "dy" + "c"_1

∴ "xy" = int "y" * "y"  "dy" + "c"_1

∴ "xy" = int "y"^2 "dy" + "c"_1

∴ "xy" = "y"^3/3 + "c"_1

∴ "y"^3/3 = "xy" + "c", where c = -c1

This is the general solution.

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