Question 22 - CBSE Class 12 Sample Paper for 2020 Boards
Last updated at Oct. 17, 2019 by Teachoo

If y = ae
^{
2x
}
+ be
^{
−x
}
, then show that (d
^{
2
}
y)/(dx
^{
2
}
) − dy/dx − 2y = 0

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Transcript

Question 22 If y = ae2x + beโx , then show that (๐^2 ๐ฆ)/(๐๐ฅ^2 ) โ ๐๐ฆ/๐๐ฅ โ 2y = 0
Given
๐ฆ=๐๐^2๐ฅ+๐๐^(โ๐ฅ)
Now,
๐๐ฆ/๐๐ฅ=2๐๐^2๐ฅโ๐๐^(โ๐ฅ)
And
(๐^2 ๐ฆ)/(๐๐ฅ^2 )=4๐๐^2๐ฅ+๐๐^(โ๐ฅ)
Now,
We need to show
(๐^2 ๐ฆ)/(๐๐ฅ^2 ) โ ๐๐ฆ/๐๐ฅ โ 2y = 0
Solving LHS
(๐^2 ๐ฆ)/(๐๐ฅ^2 ) โ ๐๐ฆ/๐๐ฅ โ 2y
= (4๐๐^2๐ฅ+๐๐^(โ๐ฅ)) โ (2๐๐^2๐ฅโ๐๐^(โ๐ฅ)) โ 2(๐๐^2๐ฅ+๐๐^(โ๐ฅ))
= 4๐๐^2๐ฅ+๐๐^(โ๐ฅ) โ 2๐๐^2๐ฅ+๐๐^(โ๐ฅ) โ 2๐๐^2๐ฅโ2๐๐^(โ๐ฅ)
= (4๐๐^2๐ฅ "โ " 2๐๐^2๐ฅ "โ " 2๐๐^2๐ฅ)+(๐๐^(โ๐ฅ) + ๐๐^(โ๐ฅ) โ2๐๐^(โ๐ฅ))
= 0 + 0
= 0
Hence proved

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