36y^{''}+108y^'+97y=0,y^'(1)=3,y(1)=4

Lösung

36 y^{^{''}} + 108 y^{^{'}} + 97 y = 0, y^{^{'}} (1) = 3, y (1) = 4

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

Schritte zur Lösung

36 y^{''} (t) + 108 y^{'} (t) + 97 y = 0

Solve linear ODE:

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

y = e^{- \frac{3 t}{2}} \frac{( 36 e ^{\frac{3}{2}} + \frac{9 e ^{\frac{3}{2}} s i n ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} - \frac{4 e ^{\frac{3}{2}} c o s ( \frac{2}{3} ) ( 27 c o s ( \frac{2}{3} ) + 8 s i n ( \frac{2}{3} ) )}{c o s ^{2} ( \frac{2}{3} ) + s i n ^{2} ( \frac{2}{3} )} ) cos ( \frac{2 t}{3} )}{2 ( - 9 cos ( \frac{2}{3} ) - 4 sin ( \frac{2}{3} ) )} + \frac{e ^{\frac{3}{2}} ( 27 cos ( \frac{2}{3} ) + 8 sin ( \frac{2}{3} ) ) sin ( \frac{2 t}{3} )}{2 ( cos ^{2} ( \frac{2}{3} ) + sin ^{2} ( \frac{2}{3} ) )}

x^{^{''}} - x^{^{'}} - 6 x = 0, x (0) = 2, x^{^{'}} (0) = - 1

y^{^{'''}} + 3 y^{^{''}} - 6 y^{^{'}} - 8 y = 0

\frac{d ^{2} y}{d x ^{2}} + 2 \frac{d y}{d x} - 15 y = 0

y^{^{''}} = 10 y

\frac{d ^{2} y}{d x ^{2}} = a^{2} y