y^{''}-y^'+y=(2e^{3t}+5t)^7

解答

y^{^{''}} - y^{^{'}} + y = (2 e^{3 t} + 5 t)^{7}

y = e^{\frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + \frac{128}{421} e^{21 t} + \frac{2240 e ^{18 t} t}{307} - \frac{78400}{94249} e^{18 t} + \frac{16800 e ^{15 t} t ^{2}}{211} - \frac{974400 e ^{15 t} t}{44521} + \frac{21168000}{9393931} e^{15 t} + \frac{10000 e ^{12 t} t ^{3}}{19} - \frac{690000 e ^{12 t} t ^{2}}{2527} + \frac{23760000 e ^{12 t} t}{336091} - \frac{362940000}{44700103} e^{12 t} + \frac{175000 e ^{9 t} t ^{4}}{73} - \frac{11900000 e ^{9 t} t ^{3}}{5329} + \frac{453600000 e ^{9 t} t ^{2}}{389017} - \frac{10210200000 e ^{9 t} t}{28398241} + \frac{107347800000}{2073071593} e^{9 t} + \frac{262500 e ^{6 t} t ^{5}}{31} - \frac{14437500 e ^{6 t} t ^{4}}{961} + \frac{472500000 e ^{6 t} t ^{3}}{29791} - \frac{10221750000 e ^{6 t} t ^{2}}{923521} + \frac{136993500000 e ^{6 t} t}{28629151} - \frac{873180000000}{887503681} e^{6 t} + 31250 t^{6} e^{3 t} - \frac{937500 e ^{3 t} t ^{5}}{7} + \frac{16875000 e ^{3 t} t ^{4}}{49} - \frac{206250000 e ^{3 t} t ^{3}}{343} + \frac{1676250000 e ^{3 t} t ^{2}}{2401} - \frac{8100000000 e ^{3 t} t}{16807} + \frac{17032500000}{117649} e^{3 t} + 78125 t^{7} + 546875 t^{6} - 16406250 t^{4} - 65625000 t^{3} + 393750000 t + 393750000

求解步骤

y^{''} (t) - y^{'} (t) + y = (2 e^{3 t} + 5 t)^{7}

求解线性方程:

y = e^{\frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + \frac{128}{421} e^{21 t} + \frac{2240 e ^{18 t} t}{307} - \frac{78400}{94249} e^{18 t} + \frac{16800 e ^{15 t} t ^{2}}{211} - \frac{974400 e ^{15 t} t}{44521} + \frac{21168000}{9393931} e^{15 t} + \frac{10000 e ^{12 t} t ^{3}}{19} - \frac{690000 e ^{12 t} t ^{2}}{2527} + \frac{23760000 e ^{12 t} t}{336091} - \frac{362940000}{44700103} e^{12 t} + \frac{175000 e ^{9 t} t ^{4}}{73} - \frac{11900000 e ^{9 t} t ^{3}}{5329} + \frac{453600000 e ^{9 t} t ^{2}}{389017} - \frac{10210200000 e ^{9 t} t}{28398241} + \frac{107347800000}{2073071593} e^{9 t} + \frac{262500 e ^{6 t} t ^{5}}{31} - \frac{14437500 e ^{6 t} t ^{4}}{961} + \frac{472500000 e ^{6 t} t ^{3}}{29791} - \frac{10221750000 e ^{6 t} t ^{2}}{923521} + \frac{136993500000 e ^{6 t} t}{28629151} - \frac{873180000000}{887503681} e^{6 t} + 31250 t^{6} e^{3 t} - \frac{937500 e ^{3 t} t ^{5}}{7} + \frac{16875000 e ^{3 t} t ^{4}}{49} - \frac{206250000 e ^{3 t} t ^{3}}{343} + \frac{1676250000 e ^{3 t} t ^{2}}{2401} - \frac{8100000000 e ^{3 t} t}{16807} + \frac{17032500000}{117649} e^{3 t} + 78125 t^{7} + 546875 t^{6} - 16406250 t^{4} - 65625000 t^{3} + 393750000 t + 393750000

y = e^{\frac{t}{2}} (c_{1} cos (\frac{3 t}{2}) + c_{2} sin (\frac{3 t}{2})) + \frac{128}{421} e^{21 t} + \frac{2240 e ^{18 t} t}{307} - \frac{78400}{94249} e^{18 t} + \frac{16800 e ^{15 t} t ^{2}}{211} - \frac{974400 e ^{15 t} t}{44521} + \frac{21168000}{9393931} e^{15 t} + \frac{10000 e ^{12 t} t ^{3}}{19} - \frac{690000 e ^{12 t} t ^{2}}{2527} + \frac{23760000 e ^{12 t} t}{336091} - \frac{362940000}{44700103} e^{12 t} + \frac{175000 e ^{9 t} t ^{4}}{73} - \frac{11900000 e ^{9 t} t ^{3}}{5329} + \frac{453600000 e ^{9 t} t ^{2}}{389017} - \frac{10210200000 e ^{9 t} t}{28398241} + \frac{107347800000}{2073071593} e^{9 t} + \frac{262500 e ^{6 t} t ^{5}}{31} - \frac{14437500 e ^{6 t} t ^{4}}{961} + \frac{472500000 e ^{6 t} t ^{3}}{29791} - \frac{10221750000 e ^{6 t} t ^{2}}{923521} + \frac{136993500000 e ^{6 t} t}{28629151} - \frac{873180000000}{887503681} e^{6 t} + 31250 t^{6} e^{3 t} - \frac{937500 e ^{3 t} t ^{5}}{7} + \frac{16875000 e ^{3 t} t ^{4}}{49} - \frac{206250000 e ^{3 t} t ^{3}}{343} + \frac{1676250000 e ^{3 t} t ^{2}}{2401} - \frac{8100000000 e ^{3 t} t}{16807} + \frac{17032500000}{117649} e^{3 t} + 78125 t^{7} + 546875 t^{6} - 16406250 t^{4} - 65625000 t^{3} + 393750000 t + 393750000

\frac{d ^{2} y}{d x ^{2}} - 9 \frac{d y}{d x} + 5 y = x e^{x}

y^{^{''}} + 4 y^{^{'}} + 4 y = e^{- t}, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} - 2 y^{^{'}} - 3 y = 3 t^{2} + 4 t - 5

y^{^{''}} - y = 7.1 cos (t), y (0) = 0, y^{^{'}} (0) = 7.1

y^{^{''}} - 4 y^{^{'}} + 3 y = 8 e^{x}