The order of the differential equation whose general
solution is
$y = C_{1} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x} + C_{4} e^{x + C_{5}}$ where $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ are arbitrary
constant is-

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Solution

$y = C_{1} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x} + C_{4} e^{x} e^{C_{5}}$

Let $C_{4} e^{C_{5}} = C_{6}$ and $C_{1} + C_{6} = C_{7}$

$\Rightarrow y = C_{7} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x}$

$\Rightarrow$ 3 arbitary constants

Order=3

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The order of the differential equation whose generalsolution is y=C1ex+C2e2x+C3e3x+C4ex+C5 where C1,C2,C3,C4,C5 are arbitraryconstant is-

y=C1ex+C2e2x+C3e3x+C4exeC5Let C4eC5=C6 and C1+C6=C7⇒y=C7ex+C2e2x+C3e3x⇒ 3 arbitary constantsOrder=3

The order of the differential equation whose general
solution is
$y = C_{1} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x} + C_{4} e^{x + C_{5}}$ where $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ are arbitrary
constant is-

$y = C_{1} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x} + C_{4} e^{x} e^{C_{5}}$

Let $C_{4} e^{C_{5}} = C_{6}$ and $C_{1} + C_{6} = C_{7}$

$\Rightarrow y = C_{7} e^{x} + C_{2} e^{2 x} + C_{3} e^{3 x}$

$\Rightarrow$ 3 arbitary constants

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