100y^{''}-80y^'+20y=0,y(3)=3,y^'(3)=8

解

100 y^{^{''}} - 80 y^{^{'}} + 20 y = 0, y (3) = 3, y^{^{'}} (3) = 8

y = e^{\frac{2 t}{5}} (\frac{( 3 ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) ) - sin ( \frac{3}{5} ) ( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) ) cos ( \frac{t}{5} )}{e ^{\frac{6}{5}} cos ( \frac{3}{5} ) ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )} + \frac{( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) sin ( \frac{t}{5} )}{e ^{\frac{6}{5}} ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )})

解答ステップ

100 y^{''} (t) - 80 y^{'} (t) + 20 y = 0

線形 ODE を解く:

y = e^{\frac{2 t}{5}} (\frac{( 3 ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) ) - sin ( \frac{3}{5} ) ( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) ) cos ( \frac{t}{5} )}{e ^{\frac{6}{5}} cos ( \frac{3}{5} ) ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )} + \frac{( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) sin ( \frac{t}{5} )}{e ^{\frac{6}{5}} ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )})

y = e^{\frac{2 t}{5}} (\frac{( 3 ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) ) - sin ( \frac{3}{5} ) ( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) ) cos ( \frac{t}{5} )}{e ^{\frac{6}{5}} cos ( \frac{3}{5} ) ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )} + \frac{( 34 cos ( \frac{3}{5} ) + 3 sin ( \frac{3}{5} ) ) sin ( \frac{t}{5} )}{e ^{\frac{6}{5}} ( cos ^{2} ( \frac{3}{5} ) + sin ^{2} ( \frac{3}{5} ) )})