1. ## [SOLVED] How to answer this question

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

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

2. Originally Posted by mark1950
Show that $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 2y\frac{dy}{dx} = 4$

given that $y = (e^{2x} - 4x + 3)^{\frac{1}{2}}$
Can you work out $\frac{dy}{dx}$?

Or $\frac{d^2y}{dx^2}$?

Can you substitute these into the DE?

3. Yes, I can but what I got was a very, very long equation.

4. Originally Posted by mark1950
Yes, I can but what I got was a very, very long equation.
Well that's how it's solved, so do it.

5. Originally Posted by mark1950
Yes, I can but what I got was a very, very long equation.
Please post your work so that it can be reviewed. Be sure to include your answer for $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.

6. @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?

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

$= \frac{e^{2x} - 2}{\sqrt{e^{2x} - 4x + 3}}$
$\frac{d^2y}{dx^2}
= \frac{(e^{2x} - 4x + 3)(2e^{2x}) - (e^{2x} - 2)^2}{\sqrt{(e^{2x} - 4x + 3)^3}}$

7. Originally Posted by mark1950
Show that $y\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 - 2y\frac{dy}{dx} = 4$

given that $y = (e^{2x} - 4x + 3)^{\frac{1}{2}}$
So now plug everything you've found into

$y\frac{d^2y}{dx^2} + \left(\frac{dy}{dx}\right)^2 - 2y\frac{dy}{dx}$

and simplify.

8. Originally Posted by mark1950
@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?

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

$= \frac{e^{2x} - 2}{\sqrt{e^{2x} - 4x + 3}}$
$\frac{d^2y}{dx^2}
= \frac{(e^{2x} - 4x + 3)(2e^{2x}) - (e^{2x} - 2)^2}{\sqrt{(e^{2x} - 4x + 3)^3}}$
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).

9. Originally Posted by mr 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).
My CAS tells me the answer is, indeed, 4.

10. Originally Posted by Prove It
My CAS tells me the answer is, indeed, 4.
It's always good to know that what you're trying to do can be done (there's always a niggling suspicion that the given solution is wrong in problems like this ....)

11. Originally Posted by mr fantastic
It's always good to know that what you're trying to do can be done (there's always a niggling suspicion that the given solution is wrong in problems like this ....)
Indubidably.

12. 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.