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)x^{2}x^{n-2} - nx.nx^{n-1}+ nx^{n}

= n(n-1)x^{n }- n^{2 }x^{n} + nx^{n}

=[n(n-1)-n^{2}+n]x^{n}

= 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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