Question

ODE problem confused!

Answer 1

Confused on b(ii)

Answer 2

I'm not sure what to do.

Answer 3

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

Answer 4

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

Answer 5

pretty similar to the part before it

Answer 6

But I will try first.

Answer 7

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

Answer 8

juts plug it in and see what happens

Answer 9

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

Answer 10

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

Answer 11

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

Answer 12

I don't know what u is.

Answer 13

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

Answer 14

Still lost...

Answer 15

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

Answer 16

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

Answer 17

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.