## ironictoaster Group Title ODE problem confused! one year ago one year ago

1. ironictoaster Group Title

2. ironictoaster Group Title

Confused on b(ii)

3. ironictoaster Group Title

I'm not sure what to do.

4. ironictoaster Group Title

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

5. Spacelimbus Group Title

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

6. colorful Group Title

pretty similar to the part before it

7. Spacelimbus Group Title

But I will try first.

8. Spacelimbus Group Title

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

9. colorful Group Title

juts plug it in and see what happens

10. experimentX Group Title
11. ironictoaster Group Title

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

12. ironictoaster Group Title

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

13. ironictoaster Group Title

I don't know what u is.

14. experimentX Group Title

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 Group Title

Still lost...

16. ironictoaster Group Title

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

17. experimentX Group Title

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

18. Spacelimbus Group Title

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 Group Title

|dw:1344360084786:dw|

20. Spacelimbus Group Title

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

21. Spacelimbus Group Title

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 Group Title

and how does that prove it equals 0?

23. ironictoaster Group Title

Yeah it pretty bad this week

24. ironictoaster Group Title

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

25. Spacelimbus Group Title

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

26. Spacelimbus Group Title

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