What is the particular solution for which y(0)=1 and y(1)=5 for the general solution of y''(t)=12t-2

Question

turingtest · Answer

how far have you gotten with this?

anonymous · Answer

I've gotten the general solution for this problem which is y=2t^3-t^2+At+B

turingtest · Answer

plug in t=0 to get B, then use the value of B and plug in t=5 to solve for A

anonymous · Answer

can you show me how its done?

turingtest · Answer

plug in t=0 into your general solution and show me what you get

anonymous · Answer

attachment

turingtest · Answer

not quite...$$y(0) = 1 = 2(0)^3-(0)^2+A(0)+B=B\implies B=?$$

anonymous · Answer

sorry i meant y(0)=2 but in this case it will be 2

turingtest · Answer

right, so now we have
$$y=2t^3-t^2+At+2$$now just do the same trick plugging in $t=1$ and solve for $A$

anonymous · Answer

a=2?

turingtest · Answer

yep :)

anonymous · Answer

so how will the equatio be written out?

turingtest · Answer

the same way you wrote it above, but with A and B replaced by the values we just found

anonymous · Answer

ok hold on

anonymous · Answer

how did you know you were solving for A and B?

turingtest · Answer

Because that's what it means to find the particular solution; particular as in if we had different initial conditions, like
y(0)=0 and y(1)=7
We would have had a different answer. Leaving the constants as variables and having no given conditions is the general solution. Filling in the constants based on a particular set of conditions gives a particular solution of the general form.

anonymous · Answer

@TuringTest can you help me with a half life question?

turingtest · Answer

if you post it separately, sure