10y^{''}+45y^'+2500y=180cos(7pix)

解

10 y^{^{''}} + 45 y^{^{'}} + 2500 y = 180 cos (7 π x)

y = e^{- \frac{9 x}{4}} (c_{1} cos (\frac{3919 x}{4}) + c_{2} sin (\frac{3919 x}{4})) + \frac{2268 π}{9604 π ^{4} - 94031 π ^{2} + 250000} sin (7 π x) + \frac{9 ( 38416 π ^{4} - 392000 π ^{2} + 1000000 )}{- 941192 π ^{6} + 14017038 π ^{4} - 71515500 π ^{2} + 125000000} cos (7 π x)

解答ステップ

10 y^{''} (x) + 45 y^{'} (x) + 2500 y = 180 cos (7 π x)

線形 ODE を解く:

y = e^{- \frac{9 x}{4}} (c_{1} cos (\frac{3919 x}{4}) + c_{2} sin (\frac{3919 x}{4})) + \frac{2268 π}{9604 π ^{4} - 94031 π ^{2} + 250000} sin (7 π x) + \frac{9 ( 38416 π ^{4} - 392000 π ^{2} + 1000000 )}{- 941192 π ^{6} + 14017038 π ^{4} - 71515500 π ^{2} + 125000000} cos (7 π x)

y = e^{- \frac{9 x}{4}} (c_{1} cos (\frac{3919 x}{4}) + c_{2} sin (\frac{3919 x}{4})) + \frac{2268 π}{9604 π ^{4} - 94031 π ^{2} + 250000} sin (7 π x) + \frac{9 ( 38416 π ^{4} - 392000 π ^{2} + 1000000 )}{- 941192 π ^{6} + 14017038 π ^{4} - 71515500 π ^{2} + 125000000} cos (7 π x)