Solve the following differential equation: dydxx2y2xydydx+x-2y2x-y=0 - Mathematics and Statistics

Sum

Solve the following differential equation:

"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0

Solution

"dy"/"dx" + ("x" - "2y")/("2x" - "y") = 0

∴ "dy"/"dx" = - (("x" - "2y")/("2x" - "y"))  ....(1)

Put y = vx

∴ "dy"/"dx" = "v + x""du"/"dx"

∴ (1) becomes, "v + x" "du"/"dx" = - (("x" - "2vx")/("2x" - "vx"))

∴ "v + x""du"/"dx" = - ((1 - "2v")/(2 - "v"))

∴ "x" "dv"/"dx" = - ((1 - "2v")/(2 - "v")) - "v"

∴ "x" "dv"/"dx" = (- 1 + "2v" - 2"v" + "v"^2)/(2 - "v")

∴ "x" "dv"/"dx" = ("v"^2 - 1)/(2 - "v")

∴ (2 - "v")/("v"^2 - 1)"dv" = 1/"x" "dx"

Integrating both sides, we get

int (2 - "v")/("v"^2 - 1)"dv" = int 1/"x" "dx"

∴ 2 int 1/("v"^2 - 1) "dv" - 1/2 int "2v"/("v"^2 - 1)"dv" = int 1/"x" "dx"

∴ 2 xx 1/2 log |("v" - 1)/("v" + 1)| - 1/2 log |"v"^2 - 1| = log |"x"| + log "c"_1   .....[because "d"/"dx" ("v"^2 - 1) = "2v" and int("f"'("x"))/("f"("x")) "dx" = log |"f"("x")| + "c"]

∴ log |("v" - 1)/("v" + 1)| - log |("v"^2 - 1)^(1/2)| = log |"c"_1 "x"|

∴ log |("v" - 1)/("v" + 1) . 1/sqrt("v"^2 - 1)| = log |"c"_1 "x"|

∴ ("v" - 1)/("v" + 1) . 1/sqrt("v"^2 - 1) = "c"_1 "x"

∴ ("y"/"x" - 1)/("y"/"x" + 1) . 1/(sqrt("y"^2/"x"^2 - 1)) = "c"_1 "x"

∴ ("y" - "x")/("y" + "x") . "x"/sqrt("y"^2 - "x"^2) = "c"_1"x"

∴ ("y" - "x")/("y" + "x") = "c"_1 sqrt("y"^2 - "x"^2)

∴ ("y" - "x")/("y" + "x") = "c"_1 sqrt(("y" - "x")("y" + "x")

∴ sqrt("y" - "x") = "c"_1 ("y" + "x")^(3/2)

∴ y – x = "c"_1^2 ("x + y")^3

∴ y – x = c(x + y)3, where c = "c"_1^2

∴ y = c(x + y)3 + x

This is the general solution.

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Chapter 6: Differential Equations - Exercise 6.4 [Page 203]
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