Question

Find the differential equation of the curve represented by xy = ae^x + be^–x + x²

Answer 1

Given xy = ae^x + 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)` = ae^x – be^–x + 2x  .......(2)

`x ("d"^2y)/("d"x^2) + ("d")/("d"x) (1) + ("d"y)/("d"x)` = ae^x + be^–x + 2 

`x ("d"^2y)/("d"x^2) + (2"d"y)/("d"x)` = ae^x + be^–x + 2  ......(3)

From (1), we get xy – x² = ae^x + 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.