The differential equation associated with the primitive , $y = c_{1} e^{3 x} + c_{2} e^{2 x} + c_{3} e^{x}$ , where $c_{1}, c_{2}, c_{3}$ are arbitrary constants be $\Rightarrow \frac{d^{3} y}{d x^{3}} - m \frac{d^{2} y}{d x^{2}} + n \frac{d y}{d x} - k y = 0$ .
Find $k + m - n$ .

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Solution

$y = c_{1} e^{3 x} + c_{2} e^{2 x} + c_{3} e^{x}$ ..... (i)
$\frac{d y}{d x} = 3 c_{1} e^{3 x} + 2 c_{2} e^{2 x} + c_{3} e^{x}$ ..... (2)
$\frac{d^{2} y}{d x^{2}} = 9 c_{1} e^{3 x} + 4 c_{2} e^{2 x} + c_{3} e^{x}$ ..... (3)
$\frac{d^{3} y}{d x^{3}} = 27 c_{1} e^{3 x} + 8 c_{2} e^{2 x} + c_{3} e^{x}$ ..... (4)
Apply $(4) 6 \times (3) + 11 \times (2) 6 \times (1)$
$\Rightarrow \frac{d^{3} y}{d x^{3}} - 6 \frac{d^{2} y}{d x^{2}} + 11 \frac{d y}{d x} - 6 y = 0$

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