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In each of the following examples, verify that the given function is a solution of the corresponding differential equation. Solution D.E. y = xn x2d2ydx2-n×xdydx+ny=0 - Mathematics and Statistics

Sum

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

Solution D.E.
y = xn `x^2(d^2y)/dx^2 - n xx (xdy)/dx + ny =0`
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Solution

y = x n

Differentiating w.r.t. x, we get

`dy/dx = nx^(n-1)`

Again, differentiating w.r.t. x, we get

`(d^2y)/dx^2 = n(n-1) x^(n-2)`

∴  `x^2(d^2y)/dx^2 - nxdy/dx +ny`

= n(n-1)x2xn-2 - nx.nxn-1+ nxn

= n(n-1)xn - n2 xn + nxn

=[n(n-1)-n2+n]xn

= 0

∴ `x^2 (d^2y)/dx^2 - nxdy/dx + ny = 0`

∴ Given function is a solution of the given differential equation.

Concept: Differential Equations
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Differential Equation and Applications
Exercise 8.1 | Q 2.2 | Page 162
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