What is the solution for y^{''}+a(dy)/(dt)+by=csin(gt) ?

The solution for y^{''}+a(dy)/(dt)+by=csin(gt) is

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y^{''}+a(dy)/(dt)+by=csin(gt)

y^{^{''}} + a \frac{d y}{d t} + b y = c sin (g t)

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{sin ( g t ) ( - c g ^{2} + c b )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}} - \frac{c g a cos ( g t )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}}

Solution steps

y^{''} (t) + a y^{'} (t) + b y (t) = c sin (g t)

Solve linear ODE:

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{sin ( g t ) ( - c g ^{2} + c b )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}} - \frac{c g a cos ( g t )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}}

y = c_{1} e^{\frac{t ( - a + a ^{2} - 4 b )}{2}} + c_{2} e^{\frac{t ( - a - a ^{2} - 4 b )}{2}} + \frac{sin ( g t ) ( - c g ^{2} + c b )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}} - \frac{c g a cos ( g t )}{g ^{2} a ^{2} + g ^{4} - 2 g ^{2} b + b ^{2}}

x^{^{''}} + 4 x^{^{'}} + 13 x = 5 cos (2 t)

y^{^{''}} - 4 y^{^{'}} + 4 y = e^{- x}

y^{^{''}} - 16 y = 3

y^{^{''}} - 3 y^{^{'}} + 2 y = 2 e^{- 3 t}, y (0) = 0, y^{^{'}} (0) = 1

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

What is the solution for y^{''}+a(dy)/(dt)+by=csin(gt) ?
The solution for y^{''}+a(dy)/(dt)+by=csin(gt) is