Form the differential equation corresponding to the function $y = A e^{x} + B e^{2 x}$ , where $A$ and $B$ are arbitrary constants.

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Solution

$y = A e^{x} + B e^{2 x}$
As the roots are $x$ and $2 x$ .
The differential equation will be-
$(D - x) (D - 2 x) = 0$
$D^{2} - 2 D x - D x + 2 x^{2} = 0$
$D^{2} - 3 D x + 2 x^{2} = 0$
$y^{''} - 3 x y^{'} + 2 x^{2} = 0 [∵ D^{n} = {y^{''}}^{. . . . n}]$
Hence the differential equation corresponding to the function $y = A e^{x} + B e^{2 x}$ is $y^{''} - 3 x y^{'} + 2 x^{2} = 0$ .

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