**Obtain the differential equation by eliminating the arbitrary constants from the following equation:**

y = c_{1}e^{2x} + c_{2}e^{5x}

#### Solution

y = c_{1}e^{2x} + c_{2}e^{5x} ....(1)

Differentiating twice w.r.t. x, we get

`"dy"/"dx" = "c"_1"e"^(2"x") xx 2 + "c"_2"e"^(5"x") xx 5`

∴ `"dy"/"dx" = 2"c"_1"e"^(2"x") + 5"c"_2"e"^(5"x")` ....(2)

and `("d"^2"y")/"dx"^2 = 2"c"_1"e"^(2"x") xx 2 + 5"c"_2"e"^(5"x") xx 5`

∴ `("d"^2"y")/"dx"^2 = 4"c"_1"e"^(2"x") + 25"c"_2"e"^("5x")` .....(3)

The equations (1), (2) and (3) are consistent in c_{1}e^{2x} and c_{2}e^{5x}

∴ determinant of their consistency is zero.

∴ `|("y",1,1),("dy"/"dx",2,5),(("d"^2"y")/"dx"^2,4,25)| = 0`

∴ y(50 - 20) - `1(25"dy"/"dx" - 5 ("d"^2"y")/"dx"^2) + 1 (4"dy"/"dx" - 2("d"^2"y")/"dx"^2) = 0`

∴ 30y - 25`"dy"/"dx" + 5("d"^2"y")/"dx"^2 + 4 "dy"/"dx" - 2("d"^2"y")/"dx"^2 = 0`

∴ `3("d"^2"y")/"dx"^2 - 21"dy"/"dx" + 30"y" = 0`

∴ `("d"^2"y")/"dx"^2 - 7"dy"/"dx" + 10"y" = 0`

This is the required D.E.

#### Notes

[**Note:** Answer in the textbook is incorrect.]