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Question
Find the differential equation of the curve represented by xy = aex + be–x + x2
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Solution
Given xy = aex + be–x + x2 ........(1)
Where a and b are aribitrary constant,
Differentiate equation (1) twice successively,
Because we have two arbitray constant.
`x ("d"y)/("d"x) + y(1)` = aex – be–x + 2x .......(2)
`x ("d"^2y)/("d"x^2) + ("d")/("d"x) (1) + ("d"y)/("d"x)` = aex + be–x + 2
`x ("d"^2y)/("d"x^2) + (2"d"y)/("d"x)` = aex + be–x + 2 ......(3)
From (1), we get xy – x2 = aex + be–x ........(4)
Substituting equation (4) in (3), we get
∴ `x ("d"^2y)/("d"x^2) + (2"d"y)/("d"x) - xy + x^2 - 2` = 0 is the required differential equation.
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