vanush
kdslkf
If (0,2) lies on the curve y = Ae^(kx) and y'' - 3(y') + 4y = 0, find the values of A and k.
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helpp (1 Viewer)
- Thread starter vanush
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Trev
stix
(0,2) satisfies y=Ae<sup>kx</sup>; so 2=Ae<sup>0</sup>; A=2.
y'=2ke<sup>kx</sup>
y"=2k<sup>2</sup>e<sup>kx</sup>
Since y"-3y'+4y=0
2k<sup>2</sup>e<sup>kx</sup>-6ke<sup>kx</sup>+8e<sup>kx</sup>=0
2e<sup>kx</sup>[k<sup>2</sup>-3k+4]=0
Since e<sup>kx</sup> can't equal zero;
When k<sup>2</sup>-3k+4=0; k=4, or, -1.
So y=2e<sup>4x</sup> or y=2e<sup>-x</sup>
y'=2ke<sup>kx</sup>
y"=2k<sup>2</sup>e<sup>kx</sup>
Since y"-3y'+4y=0
2k<sup>2</sup>e<sup>kx</sup>-6ke<sup>kx</sup>+8e<sup>kx</sup>=0
2e<sup>kx</sup>[k<sup>2</sup>-3k+4]=0
Since e<sup>kx</sup> can't equal zero;
When k<sup>2</sup>-3k+4=0; k=4, or, -1.
So y=2e<sup>4x</sup> or y=2e<sup>-x</sup>
vanush
kdslkf
ffs, i was integrating to get y'. no wonder.
thanks
thanks
vanush
kdslkf
damnit, ive spent nearly all day on this one.
anybody?
anybody?
sikeveo
back after sem2
suuuure u havnot missed somting as why is there a u in it?
vanush
kdslkf
as in units squared.
SoulSearcher
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Let b be the length of OA, and h be the length of AB
Since the area of a rectangle is base multiplied by height, Area = OA * AB
Since OA is the length of the rectangle, the area of the rectangle is equal to b * AB, but as the length of OA is x units from the point (0,0), area is x * AB
To find AB, we have to find the point at which the rectangle touches the curve y = e-x, which is when the x-co-ordinate is b.
So when x = b, y = e-b, but as x = b, y = e-x, which gives us the height of the rectanlge.
So the area of the rectangle is x * e-x = xe-x units2
Since the area of a rectangle is base multiplied by height, Area = OA * AB
Since OA is the length of the rectangle, the area of the rectangle is equal to b * AB, but as the length of OA is x units from the point (0,0), area is x * AB
To find AB, we have to find the point at which the rectangle touches the curve y = e-x, which is when the x-co-ordinate is b.
So when x = b, y = e-b, but as x = b, y = e-x, which gives us the height of the rectanlge.
So the area of the rectangle is x * e-x = xe-x units2
vanush
kdslkf
Cool. I was thinking x and the power on the exponential function (-x) were the same variable.