## ironictoaster 3 years ago ODE problem confused!

1. ironictoaster

2. ironictoaster

Confused on b(ii)

3. ironictoaster

I'm not sure what to do.

4. ironictoaster

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

5. Spacelimbus

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

6. colorful

pretty similar to the part before it

7. Spacelimbus

But I will try first.

8. Spacelimbus

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

9. colorful

juts plug it in and see what happens

10. experimentX
11. ironictoaster

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

12. ironictoaster

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

13. ironictoaster

I don't know what u is.

14. 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.

15. ironictoaster

Still lost...

16. ironictoaster

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

17. experimentX

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

18. Spacelimbus

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.

19. experimentX

|dw:1344360084786:dw|

20. Spacelimbus

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

21. Spacelimbus

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 (-:

22. ironictoaster

and how does that prove it equals 0?

23. ironictoaster

Yeah it pretty bad this week

24. ironictoaster

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

25. Spacelimbus

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

26. Spacelimbus

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