Verify that the function π¦ = π1π (βπ+2π)π₯ + π2π (βπβ2π)π₯ is a solution to π¦ β²β² + 2ππ¦ β² + (π 2 + 4)π¦ = 0.
"\ud835\udc66 = \ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65} + \ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58\u22122\ud835\udc56)\ud835\udc65} ........(1)\\\\ solution \\space to\\\\ \ud835\udc66 ^{\u2032\u2032} + 2\ud835\udc58\ud835\udc66 ^{\u2032} + (\ud835\udc58^ 2 + 4)\ud835\udc66 = 0."
"\ud835\udc66^{\u2032} = \ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58+2\ud835\udc56) + \ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58\u22122\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58-2\ud835\udc56)\\\\\ud835\udc66^{\u2032\u2032} = \ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58+2\ud835\udc56)^2 + \ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58\u22122\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58-2\ud835\udc56)^2"
"\ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58+2\ud835\udc56)^2 + \ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58\u22122\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58-2\ud835\udc56)^2+\n\n\n2k(\ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58+2\ud835\udc56)+\n\n\n\ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58-2\ud835\udc56)\ud835\udc65}(\u2212\ud835\udc58+2\ud835\udc56)+(k^2+4)\ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}+\ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58-2\ud835\udc56)}"
"\ud835\udc50_1 (\u2212\ud835\udc58+2\ud835\udc56)^2 \ud835\udc52^{ (\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65}+\n\n\ud835\udc50_2\ud835\udc52 ^{(\u2212\ud835\udc58-2\ud835\udc56)\ud835\udc65)}(-k-2i)^2 + 2k(\u2212\ud835\udc58\u22122\ud835\udc56)\n\n\ud835\udc50_1\ud835\udc52 ^{(\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65 }+ 2k (\u2212\ud835\udc58\u22122\ud835\udc56)\n\n\n\ud835\udc50_2\ud835\udc52^{ (\u2212\ud835\udc58-2\ud835\udc56)\ud835\udc65} + (\ud835\udc58^2+4)\n\n\ud835\udc50_1\ud835\udc52^{ (\u2212\ud835\udc58+2\ud835\udc56)\ud835\udc65 }+ (\ud835\udc58^2+4)\n\n\n\ud835\udc50_2\ud835\udc52^ {(\u2212\ud835\udc58+2\ud835\udc56)k}"
hence verified
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