Form the differential equation by eliminating arbitrary constants from the relation $y = A e^{5 x} + B e^{- 5 x}$ .

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Solution

Fact: $\frac{d}{d x} (e^{a x}) = a e^{a x}$ , where $a$ is constant

Now given $y = A e^{5 x} + B e^{- 5 x}$ ,........(I)
Here there are two arbitrary constants, so we need to differentiate it twice
Now differentiate (1)
$y^{'} = 5 A e^{5 x} - 5 B e^{- 5 x}$ .........(II)
Now 5(I)+(II) $\Rightarrow 5 y + y^{'} = 10 A e^{5 x}$ .....(III)

Differentiate (III) again
$5 y^{'} + y^{''} = 50 A e^{5 x}$
$\Rightarrow 5 y^{'} + y^{''} = 5 (10 A e^{5 x}) = 10 (5 y + y^{'})$ , using (III)
$\Rightarrow y^{''} - 5 y^{'} - 50 y = 0$

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