Obtain 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,
Find the order of the differential equation

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Solution

The correct option is C 3
$y = c_{1} e^{3 x} + c_{2} e^{2 x} + c_{3} e^{x}$
$\frac{d y}{d x} = 3 c_{1} e^{3 x} + 2 c_{2} e^{2 x} + c_{1} e^{x}$ ..(i)
$\frac{d^{2} y}{d x^{2}} = 9 c_{1} e^{3 x} + 4 c_{2} e^{2 x} + c_{1} e^{x}$ ..(ii)
$\frac{d^{3} y}{d x^{3}} = 27 c_{1} e^{3 x} + 8 c_{2} e^{2 x} + c_{1} e^{x}$ ..(iii)
Solving (i), (ii), (iii) we get
$\frac{d^{3} y}{d x^{3}} - 6 \frac{d^{2} y}{d x^{2}} + 11 \frac{d y}{d x} - 6 y = 0$
Clearly order of this differential equation is 3.

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