y^{''}+4y=\H(x-pi)-\H(x-2pi)

Lösung

y^{^{''}} + 4 y = H (x - π) - H (x - 2 π)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x + sin (2 x) \cdot \int \frac{cos ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x

Schritte zur Lösung

y^{''} (x) + 4 y = H (x - π) - H (x - 2 π)

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x + sin (2 x) \cdot \int \frac{cos ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x

y = c_{1} cos (2 x) + c_{2} sin (2 x) + cos (2 x) \cdot \int - \frac{sin ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x + sin (2 x) \cdot \int \frac{cos ( 2 x ) ( H ( x - π ) - H ( x - 2 π ) )}{2} d x

y^{^{''}} + 2 y^{^{'}} + y = 1 - H (t - 1)

y^{^{''}} + 4 y = 8 t^{2}

y^{^{''}} - y = sin (x) + e^{x}

y^{^{''}} + y = t^{2}

5 y^{^{''}} + 6 y^{^{'}} + 80 y = - 24 sin (4 t)