81y^{''}-18y^'+5y=0,y(1)=1,y^'(1)=7

Lösung

81 y^{^{''}} - 18 y^{^{'}} + 5 y = 0, y (1) = 1, y^{^{'}} (1) = 7

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}

Schritte zur Lösung

81 y^{''} (t) - 18 y^{'} (t) + 5 y = 0

Solve linear ODE:

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}

y = e^{\frac{t}{9}} \frac{( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) - 9 e sin ( \frac{2}{9} ) ( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) ) cos ( \frac{2 t}{9} )}{9 e cos ( \frac{2}{9} ) ( - 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} ) )} + \frac{( 62 cos ( \frac{2}{9} ) + 2 sin ( \frac{2}{9} ) ) sin ( \frac{2 t}{9} )}{- 9 e cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e sin ^{2} ( \frac{2}{9} ) + e ^{\frac{1}{9}} cos ( \frac{2}{9} ) sin ( \frac{2}{9} ) + 2 9 e cos ^{2} ( \frac{2}{9} )}

9 x^{^{''}} + 9 x^{^{'}} - 4 x = 0

y^{^{''}} + y = 0, y (π) = 1, y (0) = 1

a \cdot y^{^{''}} + b \cdot y = 0

y^{^{''}} - 8 y^{^{'}} + 16 y = 0, y (0) = 4, y^{^{'}} (0) = 4

y^{^{''}} + 8 y^{^{'}} + 15 y = 0, y (0) = 12, y^{^{'}} (0) = - 52