Question

(d^2 y)/〖dx〗^2 +dy/dx+y=e^2x​

Answer 1

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$solve \: \frac{ {d}^{2}y }{ {dx}^{2} } + 2 \frac{dy}{dx} + y = {e}^{2x} \\ its \: symbolic \: form \: is \: ( {D}^{2} + 2D + 1)y = {e}^{2x} \\ so \: auxillary \: equation \: is \: \\ {D}^{2} + 2D + 1 = 0 \\ {(D + 1)}^{2} = 0 \\ = > \: D = - 1, - 1 \\ so \: C.F. \: = (a + bx) {e}^{ - x}$

$Particular Integral, P.I. = \frac{1}{ {(D + 1)}^{2} } {e}^{2x} \\ = \frac{1}{ {(2 + 1)}^{2} } {e}^{2x} \\ = \frac{1}{9} {e}^{2x}$

$\large\bold\red{Complete solution = C.F. + P.I.}$

$\small\bold\red{ = (a + bx) {e}^{ - x} + \frac{1}{9} {e}^{2x} }$

$\huge \fcolorbox{black}{cyan}{♛hope \: it \: helps \: you♛}$