Find the differential equation corresponding to y = ae^2x + be⁻^3x + ce^x where a, b, c are arbitrary constants.

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Solution

We have,
y = ae^2x + be⁻^3x + ce^x .....(1)
Differentiating with respect to x, we get
$\frac{d y}{d x} = 2 a e^{2 x} - 3 b e^{- 3 x} + c e^{x} . . . . . (2) \Rightarrow \frac{d^{2} y}{d x^{2}} = 4 a e^{2 x} + 9 b e^{- 3 x} + c e^{x} \Rightarrow \frac{d^{3} y}{d x^{3}} = 8 a e^{2 x} - 27 b e^{- 3 x} + c e^{x} \Rightarrow \frac{d^{3} y}{d x^{3}} = 7 (2 a e^{2 x} - 3 b e^{- 3 x} + c e^{x}) - 6 (a e^{2 x} + b e^{- 3 x} + c e^{x}) \Rightarrow \frac{d^{3} y}{d x^{3}} = 7 (\frac{d y}{d x}) - 6 y [Using (1) and (2)] \Rightarrow \frac{d^{3} y}{d x^{3}} - 7 (\frac{d y}{d x}) + 6 y = 0$

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Similar questions

y = {a e}^{2 x} + {b e}^{- 3 x} + {c e}^{x}

y = a e^{2 x} + b e^{- 3 x}

y = A e^{x} + B e^{2 x}

A

B

y = a e^{2 x} + b e^{- 2 x}

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