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DE Q, different correct methods - different answers ?

\displaystyle \dfrac{dy}{dx} +2y = e^x

Method of the integrating factor:

\displaystyle I = e^{2x}

Unparseable latex formula:

\displaystyle \Rightarrow e^{2x}\dfrac{dy}{dx} + 2ye^{2x} = e^{4x}[br][br]\Rightarrow \dfrac{d}{dx} (ye^{2x}) = e^4x[br]\[br]\Rightarrow y = \dfrac{e^{2x}}{4} + Ce^{-2x}

To the best of my knowledge there aren't any mistakes in there.

Now the complementary function - particular integral method:

Reduced equation:

Unparseable latex formula:

\displaystyle \dfrac{dy}{dx} = -2y [br]\[br]y_c = Ce^{-2x}

Going back to the complete function now and finding a particular solution:

let

Unparseable latex formula:

\displaystyle y = Ae^x [br]\[br] [br]\dfrac{dy}{dx} +2y = e^x \Rightarrow Ae^x + 2Ae^x = e^{2x} \Rightarrow A=\dfrac{1}{3}

Hence:

\displaystyle y_p = \dfrac{e^x}{3}

The general solution:

\displaystyle y = y_c + y_p \Rightarrow y = Ce^{-2x} + \dfrac{e^x}{3}

I can't see where I have gone wrong on either so why is there a difference ? Are there two general solutions ?

Thnx

(edited 11 years ago)

Reply 1

ghostwalker

Original post by member910132

\displaystyle I = e^{2x}

\displaystyle \Rightarrow e^{2x}\dfrac{dy}{dx} + 2ye^{2x} = e^{4x}[br][br]

RHS of 2nd quoted line should be

e^{3x}

Not checked the rest.

Reply 2

IxI_Rhys_IxI

On the Integrating Factor, shouldn't e^x * e^2x become e^3x by the laws of indicies?

Reply 3

Intriguing Alias

Original post by member910132

\displaystyle \dfrac{dy}{dx} +2y = e^x

Method of the integrating factor:

\displaystyle I = e^{2x}

Unparseable latex formula:

\displaystyle \Rightarrow e^{2x}\dfrac{dy}{dx} + 2ye^{2x} = e^{4x}[br][br]\Rightarrow \dfrac{d}{dx} (ye^{2x}) = e^4x[br]\[br]\Rightarrow y = \dfrac{e^{2x}}{4} + Ce^{-2x}