Question

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

• A. 5
• B. 4
• C. 3
• D. None of these

Answer 1

Answer: (C)

3

Answer 2

$\mathbf{y =}\mathbf{c}_{\mathbf{1}}\mathbf{e}^{\mathbf{x}}\mathbf{+}\mathbf{c}_{\mathbf{2}}\mathbf{e}^{\mathbf{2x}}\mathbf{+}\mathbf{c}_{\mathbf{3}}\mathbf{e}^{\mathbf{3x}}\mathbf{+}\mathbf{c}_{\mathbf{4}}\mathbf{e}^{\mathbf{x}}\mathbf{.}\mathbf{e}^{\mathbf{c}_{\mathbf{5}}}$

$\mathbf{= (}\mathbf{c}_{\mathbf{1}}\mathbf{+}\mathbf{c}_{\mathbf{4}}\mathbf{.}\mathbf{e}^{\mathbf{c}_{\mathbf{5}}}\mathbf{)}\mathbf{e}^{\mathbf{x}}\mathbf{+}\mathbf{c}_{\mathbf{2}}\mathbf{e}^{\mathbf{2x}}\mathbf{+}\mathbf{c}_{\mathbf{3}}\mathbf{e}^{\mathbf{3x}}\mathbf{=}\mathbf{c}_{\mathbf{1}}^{\mathbf{'}}\mathbf{e}^{\mathbf{x}}\mathbf{+}\mathbf{c}_{\mathbf{2}}\mathbf{e}^{\mathbf{2x}}\mathbf{+}\mathbf{c}_{\mathbf{3}}\mathbf{e}^{\mathbf{3x}}$

where $c_{1}^{'} = c_{1} + c_{4}.e^{c_{5}}$. So there are 3 arbitrary constant associated with different terms. Hence the order of the differential equation formed, will be 3.