Show that $\displaystyle y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 2y\frac{dy}{dx} = 4 $

given that $\displaystyle y = (e^{2x} - 4x + 3)^{\frac{1}{2}}$

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- Sep 24th 2009, 10:53 PMmark1950[SOLVED] How to answer this question
Show that $\displaystyle y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 2y\frac{dy}{dx} = 4 $

given that $\displaystyle y = (e^{2x} - 4x + 3)^{\frac{1}{2}}$ - Sep 24th 2009, 10:55 PMProve It
- Sep 25th 2009, 12:34 AMmark1950
Yes, I can but what I got was a very, very long equation.

- Sep 25th 2009, 12:35 AMProve It
- Sep 25th 2009, 12:57 AMmr fantastic
- Sep 25th 2009, 01:32 AMmark1950
@Prove it

Dude, if I knew how to solve it, I would not have posted this question here.

@Mr. Fantastic

Thanks. After this step, I simply do not know how to continue. I tried substituting dy/dx into them but simply couldn't get rid of the exponential, e so as to prove that its right. You know what I mean, right?

$\displaystyle \frac{dy}{dx}

= \frac{1}{2}(e^{2x} - 4x + 3)^{-\frac{1}{2}}(2e^{2x} - 4) $

$\displaystyle = \frac{e^{2x} - 2}{\sqrt{e^{2x} - 4x + 3}}$

$\displaystyle \frac{d^2y}{dx^2}

= \frac{(e^{2x} - 4x + 3)(2e^{2x}) - (e^{2x} - 2)^2}{\sqrt{(e^{2x} - 4x + 3)^3}}$ - Sep 25th 2009, 01:39 AMProve It
- Sep 25th 2009, 01:48 AMmr fantastic
Your derivatives are OK. Now the unfortunate fact is that you need to substitute y and the derivatives into the left hand side of the differential equation. Then simplify it (hopefully to 4). This will take a fair bit of algebra but is not difficult - just very tedious (hence my use above of the word unfortunate).

- Sep 25th 2009, 01:49 AMProve It
- Sep 25th 2009, 01:53 AMmr fantastic
- Sep 25th 2009, 01:54 AMProve It
- Sep 25th 2009, 04:10 AMmark1950
Oh darn...I was looking for a simpler way but nvm...there are some things that can't be simplified. Thanks, anyway for the assurance that my derivatives are correct.