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Integral Calculus

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Joined:	16 Jan 2009
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8 May 2009 13:47:44 IST

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DIFFERENTIAL EQUATION PROBLEM

None

Differential equation corresponding to

y=a1e^(m1x)+a2e^(m2x)+a3e^(m3x)

a1,a2,a3 are arbitary constants and m1 , m2 m3 are roots of m^3-2m^2+4=0-->eqn1

SOLUTION-->States that substitute m by dy/dx in eqn1.I dont get it.

ANy one explain?

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RyuAmakusa

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Joined: 21 Mar 2008

Posts: 776

16 May 2009 01:24:53 IST

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$ext{Well...at first i did not realise what "substitute m by dy/dx in eqn1" meant.....} \ ext{but i solved it, after seeing the ans.....i realised something.......i$

$\\ ext{There is something called a differential operator D }\\ D= rac{d}{dx} ext{ and }D^2= rac{d^2}{dx^2}......\\ ext{This they use for solving diff. eq say for the above ans. they sub. } D ; or ; m= rac{d}{dx} ext{ so u get the characterestic eq. as}\\ m^3-2m^2+4=0\\ ext{if a characterstic eq. has n real and distinct roots }m_1,m_2,....m_n ext{ then the sol is of the form }\\a_1e^{m_1x}+a_2e^{m_2x}+.....a_ne^{m_nx}\\ ext{if it has equal roots then there is another form.....there are 2 or 3 things like this}\ ext{so now the question is the sol of the diff eq corresponding to the above characterstic eq. }\ ext{so the diff. eq for the question is the diff. eq corresponding to the above characterstic eq.}\ ext{which can be obtained by sub. }m = rac{d}{dx} ext{ in fact it is } my= rac{dy}{dx}\ ext{and u will get the same ans as the above method ie. }y$

is; every; thing; clear; if; not; u; can; ask; me......

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