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May 28th, 2017, 04:35 AM   #1
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Higher order D.E

The general solution of $y'''-4y''+y'=0$ is

A) $c_1 \sinh^2x+c_2\cosh^2x+c_3$
B) $c_1 \sinh2x+c_2 \cosh2x+c_3$
C) $c_1 \sin2x+c_2 \cos2x+c_3$
D) $c_1e^{2x}+c_2e^{-2x}+c_3$

I got the roots as follows : $0$,$\displaystyle 2\pm\sqrt{3}$

How should I include these roots in the general solution format and which one is it?

Please help!
Thank you.

Last edited by skipjack; May 28th, 2017 at 06:16 PM.
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May 28th, 2017, 08:08 AM   #2
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The solution is found in the form $\displaystyle C\cdot e^{\lambda x}, C\in \mathbb{R}$.
One substitutes it for $\displaystyle y$:

$\displaystyle (C\cdot e^{\lambda x})''' -4(C\cdot e^{\lambda x})''+ (C\cdot e^{\lambda x})' =0.$
$\displaystyle (\lambda^3-4\lambda^2+\lambda)C\cdot e^{\lambda x}=0.$

So,
$\displaystyle y(x)=C_1\cdot e^{\lambda_1 x} +C_2\cdot e^{\lambda_2 x}+C_3\cdot e^{\lambda_3 x}.$

Last edited by ABVictor; May 28th, 2017 at 08:11 AM.
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May 28th, 2017, 11:26 AM   #3
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The characteristic equation is . We have r = 0 and so or or . That means that the general solution to the differential equation is .

NONE of the given possibilities is correct.

Last edited by skipjack; May 28th, 2017 at 06:46 PM.
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May 28th, 2017, 05:52 PM   #4
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Quote:
Originally Posted by Country Boy View Post
The characteristic equation is . We have r = 0 and so or or . That means that the general solution to the differential equation is .

NONE of the given possibilities is correct.
How could it be
It is a Post graduation Entrance paper and moreover this question can have multiple answers!

Please check if any other possible way to get the general solution

Last edited by skipjack; May 28th, 2017 at 06:46 PM.
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May 28th, 2017, 06:23 PM   #5
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Quote:
Originally Posted by Lalitha183 View Post
How could it be
It is a Post graduation Entrance paper and moreover this question can have multiple answers!

Please check if any other possible way to get the general solution
Did you read post #2?
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May 28th, 2017, 06:43 PM   #6
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Obviously, "$t$" should be "$x$". Are you sure you typed the original equation correctly, Lalitha183?
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May 28th, 2017, 07:09 PM   #7
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Quote:
Originally Posted by skipjack View Post
Obviously, "$t$" should be "$x$". Are you sure you typed the original equation correctly, Lalitha183?
Yes, I did.

I have read the post #2...So that could leave option D as the answer ?
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May 28th, 2017, 09:02 PM   #8
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Quote:
Originally Posted by Joppy View Post
Did you read post #2?
I have read it. But the option doesn't seems to be equivalent to the answer that came up.

Last edited by skipjack; May 28th, 2017 at 09:49 PM.
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May 28th, 2017, 09:12 PM   #9
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Originally Posted by Lalitha183 View Post
I have read it. But the option doesn't seems to be equivalent to the answer that came up.
Take a look at the approximate form to the solution here.

Last edited by skipjack; May 28th, 2017 at 09:49 PM.
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May 28th, 2017, 09:15 PM   #10
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Quote:
Originally Posted by Joppy View Post
Take a look at the approximate form to the solution here.
That link is not working it seems.

Last edited by skipjack; May 28th, 2017 at 09:47 PM.
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