Solve the following differential equation: dydx2yx2y + xdydx=2y-x2y + x - Mathematics and Statistics

Sum

Solve the following differential equation:

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

Solution

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

Put y = vx      ∴ "dy"/"dx" = "v + x""dv"/"dx"

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

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

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

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

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

Integrating both sides, we get

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

∴ int (1/2 ("4v" - 1) + 3/2)/("2v"^2 - "v" + 1) "dv" = - int 1/"x" "dx"

∴ 1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/2 int 1/("2v"^2 - "v" + 1) "dv" = - int 1/"x"

∴ 1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4 int 1/("v"^2 - 1/2"v" + 1/2)"dv" = - int 1/"x" "dx"

∴ 1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4 int 1/(("v"^2 - 1/2"v" + 1/16) + 7/16) "dv" = - int 1/"x" "dx"

∴ 1/2 int ("4v" - 1)/("2v"^2 - "v" + 1) "dv" + 3/4int 1/(("v" - 1/4)^2 + (sqrt7/4)^2)"dv" = - int 1/"x" "dx"

∴ 1/2 log |2"v"^2 - "v" + 1| + 3/4 xx 1/((sqrt7/4)) tan^-1 |("v" - 1/4)/((sqrt7/4))| = - log |x| + "c"_1 .....[because "d"/"dv" (2"v"^2 - "v" + 1) = 4"v" - 1 and int ("f"'("v"))/("f"("v")) "dv" = log |"f"("v")| + c]

∴ 1/2 log |2 ("y"^2/"x"^2) - "y"/"x" + 1| + 3/sqrt7 tan^-1 ((4("y"/"x") - 1)/sqrt7) = - log |"x"| + "c"_1

∴ 1/2 log |(2"y"^2 - "xy" + "x"^2)/"x"^2| + 3/sqrt7 tan^-1 ((4"y - x")/(sqrt7"x")) = - log |"x"| + "c"_1

∴ log |("x"^2 - "xy" + "2y"^2)/"x"^2| + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = - 2 log |"x"| + 2"c"_1

∴ log |"x"^2 - "xy" + "2y"^2| - log"x"^2 + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = - log "x"^2 + "c"_1  "where"  "c" = 2"c"_1

∴ log |"x"^2 - "xy" + "2y"^2| + 6/sqrt7 tan^-1 (("4y - x")/(sqrt7"x")) = "c"

This is the general solution.

Notes

[Note: Answer in the textbook is incorrect.]

Concept: Formation of Differential Equations
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Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 6 Differential Equations
Miscellaneous exercise 6 | Q 5.3 | Page 217