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Question
Find the differential equation corresponding to y = ae2x + be−3x + cex where a, b, c are arbitrary constants.
Solution
We have,
y = ae2x + be−3x + cex .........(1)
Differentiating with respect to x, we get
\[\frac{dy}{dx} = 2a e^{2x} - 3b e^{- 3x} + c e^x . . . . . . . . \left( 2 \right)\]
\[ \Rightarrow \frac{d^2 y}{d x^2} = 4a e^{2x} + 9b e^{- 3x} + c e^x \]
\[ \Rightarrow \frac{d^3 y}{d x^3} = 8a e^{2x} - 27b e^{- 3x} + c e^x \]
\[ \Rightarrow \frac{d^3 y}{d x^3} = 7\left( 2a e^{2x} - 3b e^{- 3x} + c e^x \right) - 6\left( a e^{2x} + b e^{- 3x} + c e^x \right)\]
\[ \Rightarrow \frac{d^3 y}{d x^3} = 7\left( \frac{dy}{dx} \right) - 6y ...........\left[\text{Using (1) and (2)} \right]\]
\[ \Rightarrow \frac{d^3 y}{d x^3} - 7\left( \frac{dy}{dx} \right) + 6y = 0\]
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