Question

Find the differential equation by eliminating arbitrary constants from the relation y = (c₁ + c₂x)e^x 

Answer 1

y = (c₁ + c₂x)e^x       ......(i)

Here, c₁ and c₂ are arbitrary constants.

Differentiating w.r.t. x, we get

`("d"y)/("d"x)` = (c₁ + c₂x)e^x + c₂e^x

∴ `("d"y)/("d"x)` = y + c₂e^x   ......(ii) .......[From(i)]

Again, differentiating w.r.t. x, we get

`("d"^2y)/("d"x^2) = ("d"y)/("d"x) + "c"_2"e"^x`

∴ c₂e^x = `("d"^2y)/("d"x^2) - ("d"y)/("d"x)`   .....(iii)

Substituting (iii) in (ii), we get

`("d"y)/("d"x) = y + ("d"^2y)/("d"x^2) -  ("d"y)/("d"x)`

∴ `("d"^2y)/("d"x^2) - 2 ("d"y)/("d"x) + y` = 0