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In the following example verify that the given function is a solution of the differential equation. x2yyxyxydxdyx2=2y2logy, x2+y2=xydxdy - Mathematics and Statistics

Sum

In the following example verify that the given function is a solution of the differential equation.

`"x"^2 = "2y"^2 log "y",  "x"^2 + "y"^2 = "xy" "dx"/"dy"`

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Solution

`"x"^2 = "2y"^2 log "y"`      .....(1)

Differentiating both sides w.r.t. y, we get

`"2x" "dx"/"dy" = 2 "d"/"dy" ("y"^2 log "y")`

`= 2 ["y"^2 "d"/"dy" (log "y") + (log "y") * "d"/"dy"("y"^2)]`

`= 2 ["y"^2 xx 1/"y" + (log "y") xx "2y"]`

∴ `"x" "dx"/"dy" = "y" + 2"y" log "y"`

∴ `"xy" "dx"/"dy" = "y"^2 + 2"y"^2 log "y"`

= y2 + x2       ....[By (1)]

∴ `"x"^2 + "y"^2 = "xy" "dx"/"dy"`

Hence, x2 = 2y2 log y is a solution of the D.E.

`"x"^2 + "y"^2 = "xy" "dx"/"dy"`

Concept: Formation of Differential Equations
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 6 Differential Equations
Miscellaneous exercise 6 | Q 2.5 | Page 217
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