The P.I. of $(3 D^{2} + D - 14) y = 13 e^{2 x}$ is:

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Solution

A particular solution (P.I. ) of the differential equation.
$P . I . = \frac{1}{f (D)} \cdot 13 e^{2 x} \Rightarrow \frac{1}{3 D^{2} + D - 14} \cdot 13 e^{2 x} \Rightarrow \frac{1}{3 {(2)}^{2} + 2 - 14} \cdot 13 e^{2 x} [Replace D by 2] \Rightarrow \frac{1}{12 + 2 - 14} \cdot 13 e^{2 x} \Rightarrow \frac{1}{0} \cdot 13 e^{2 x} ∵ 3 D^{2} + D - 14 = 0 ∴ P . I . = \frac{1}{f' (D)} \cdot 13 e^{2 x} \cdot x \Rightarrow \frac{1}{6 D + 1} \cdot 13 x e^{2 x} \Rightarrow \frac{1}{6 (2) + 1} \cdot 13 x e^{2 x} \Rightarrow \frac{1}{13} \cdot 13 x e^{2 x} \Rightarrow x e^{2 x}$
Hence, the P.I. of the differential equation is $x e^{2 x}$ .

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The P.I. of (3D2+D−14)y=13e2x is:

The P.I. of $(3 D^{2} + D - 14) y = 13 e^{2 x}$ is: