- anonymous

ODE problem confused!

- jamiebookeater

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- anonymous

##### 1 Attachment

- anonymous

Confused on b(ii)

- anonymous

I'm not sure what to do.

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## More answers

- anonymous

I have to find u somehow I reckon, not 100% though.

- anonymous

Hmm I believe you just have to show that the identity is valid, because the roots are identical.

- anonymous

pretty similar to the part before it

- anonymous

But I will try first.

- anonymous

the other solution will have an additional x infront of it I am not mistaken.

- anonymous

juts plug it in and see what happens

- experimentX

http://en.wikipedia.org/wiki/Reduction_of_order

- anonymous

If it's double roots then yes there's extra x in the solution.

- anonymous

I'm still not sure what should I do first.

- anonymous

I don't know what u is.

- experimentX

assume that \( x(t) = u(t) x_1(t) \) is another solution ... find the value of u(t) ... so that you have complete solution.

- anonymous

Still lost...

- anonymous

@experimentX I genuinely don't know where to start.

- experimentX

i think there is an example in the wikipedia ... in the link i posted above.

- anonymous

If I understand this problem then they just want you to check what happens if you substitute back their provided result. I believe their are trying to introduce you to the method of Reduction of Order
You will get a result in the form of
\[ \Large u(x)=d_1x+d_2 \]
where \(d_1, d_2\) are constant. The second solution is of the form
\[ \Large y_2(x)=u(x)e^{\frac{x}{2}}\]
So you can use superposition to get the general solution.

- experimentX

|dw:1344360084786:dw|

- anonymous

Note that \[ \Large 4r^2-4r+1=0 \] Has a discriminant of zero.

- anonymous

Pardon me if I was interrupting something in here, OpenStudy lags horribly for me today so I hit the post button before it crashes me again (-:

- anonymous

and how does that prove it equals 0?

- anonymous

Yeah it pretty bad this week

- anonymous

What I understand so far, we need to sub ux into equation 4

- anonymous

\[4\left( u''e^{\frac{t}{2}} + \frac{1}{2}e^{\frac{t}{2}}u' + \frac{1}{4}e^{\frac{t}{2}}u + \frac{1}{2}u'e^{\frac{t}{2}}\right) -4 \left(u' e^{\frac{t}{2}}+ \frac{1}{2}e^{\frac{t}{2}}u \right)+ ue^{\frac{t}{2}}=0\]

- anonymous

Divide by \( \large e^{\frac{t}{2}} \) and then see what happens when you bring it all together.

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