y^{''}+4y=cos(x)cos(3x)

Lösung

y^{^{''}} + 4 y = cos (x) cos (3 x)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) cos ( x ) cos ( 3 x )}{2} d x + \frac{1}{2} (\frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x)))) sin (2 x)

Schritte zur Lösung

y^{''} (x) + 4 y = cos (x) cos (3 x)

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) cos ( x ) cos ( 3 x )}{2} d x + \frac{1}{2} (\frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x)))) sin (2 x)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) cos ( x ) cos ( 3 x )}{2} d x + \frac{1}{2} (\frac{1}{2} cos (x) (sin (x) + \frac{1}{5} sin (5 x)) + \frac{1}{2} (\frac{1}{2} (x - \frac{1}{2} sin (2 x)) + \frac{1}{10} (\frac{1}{4} sin (4 x) - \frac{1}{6} sin (6 x)))) sin (2 x)

y^{^{''}} + 3 y = - 48 t^{2} e^{3 t}

so l v e f or d, d (t) = \frac{100 ln ( t + 2 )}{t + 2}

y^{^{''}} + 3 y^{^{'}} = 1

y^{^{''}} + y = 2 x

y^{^{''}} - y^{^{'}} = t e^{t}, y (0) = 2, y^{^{'}} (0) = 1