Question

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

Answer 1

We have,

y = ae^2x + be^−3x + ce^x            .........(1)

Differentiating with respect to x, we get

\[\frac{dy}{dx} = 2a e^{2x} - 3b e^{- 3x} + c e^x . . . . . . . . \left( 2 \right)\]

\[ \Rightarrow \frac{d^2 y}{d x^2} = 4a e^{2x} + 9b e^{- 3x} + c e^x \]

\[ \Rightarrow \frac{d^3 y}{d x^3} = 8a e^{2x} - 27b e^{- 3x} + c e^x \]

\[ \Rightarrow \frac{d^3 y}{d x^3} = 7\left( 2a e^{2x} - 3b e^{- 3x} + c e^x \right) - 6\left( a e^{2x} + b e^{- 3x} + c e^x \right)\]

\[ \Rightarrow \frac{d^3 y}{d x^3} = 7\left( \frac{dy}{dx} \right) - 6y ...........\left[\text{Using (1) and (2)} \right]\]

\[ \Rightarrow \frac{d^3 y}{d x^3} - 7\left( \frac{dy}{dx} \right) + 6y = 0\]