# In the following example verify that the given expression is a solution of the corresponding differential equation: y = xm; xdydxmxdydxmyx2d2ydx2-mxdydx+my=0 - Mathematics and Statistics

Sum

In the following example verify that the given expression is a solution of the corresponding differential equation:

y = xm; "x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0

#### Solution

y = x

Differentiating twice w.r.t. x, we get

"dy"/"dx" = "d"/"dx" ("x"^"m") = "mx"^("m - 1")

and ("d"^2"y")/"dx"^2 = "d"/"dx" ("mx"^("m - 1")) = "m" "d"/"dx" ("x"^("m - 1")) = "m"("m" - 1) "x"^("m - 2")

∴ "x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my"

= "x"^2 * "m"("m" - 1) "x"^("m - 2") - "mx" * "mx"^("m" - 1) + "m" * "x"^"m"

= "m"("m - 1") "x"^"m" - "m"^2 "x"^"m" + "mx"^"m"

= ("m"^2 - "m" - "m"^2 + "m")"x"^"m" = 0

This shows that y = xm is a solution of the D.E.

"x"^2 ("d"^2"y")/"dx"^2 - "mx" "dy"/"dx" + "my" = 0.

Concept: Formation of Differential Equations
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