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DEtools[matrix_riccati] - solve a Matrix Riccati differential equation
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Calling Sequence
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matrix_riccati(A, K, Z0, t)
matrix_riccati(A, K, Z0, t=t0)
matrix_riccati(A, K, t)
matrix_riccati(A, K, t=t0)
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Parameters
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A, K
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Matrix coefficients of the Riccati Matrix Equation
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Z0
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Matrix representing the initial conditions for t=t0
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t
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independent variable
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t0
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position of initial conditions
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Description
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The Matrix Riccati differential equation is
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The command with(DEtools,matrix_riccati) allows the use of the abbreviated form of this command.
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Examples
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The unknowns are 
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![Z := Matrix([[x(t), y(t)], [y(t), -x(t)]])](/support/helpjp/helpview.aspx?si=8699/file01351/math136.png)
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![Z := Matrix([[x(t), y(t)], [y(t), -x(t)]])](/support/helpjp/helpview.aspx?si=8699/file01351/math139.png)
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A simple example would be
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![A := Matrix([[-t, 1], [1, t]])](/support/helpjp/helpview.aspx?si=8699/file01351/math145.png)
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![A := Matrix([[-t, 1], [1, t]])](/support/helpjp/helpview.aspx?si=8699/file01351/math148.png)
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![K := Matrix([[c, 0], [0, c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math152.png)
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![K := Matrix([[c, 0], [0, c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math155.png)
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The matrix with the initial values ![x(t[0])](/support/helpjp/helpview.aspx?si=8699/file01351/math158.png) and ![y(t[0])](/support/helpjp/helpview.aspx?si=8699/file01351/math160.png) is
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![Z0 := Matrix([[_C1, _C2], [_C2, -_C1]])](/support/helpjp/helpview.aspx?si=8699/file01351/math165.png)
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![Z0 := Matrix([[_C1, _C2], [_C2, -_C1]])](/support/helpjp/helpview.aspx?si=8699/file01351/math168.png)
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The system of equations represented by these matrices is thus
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![sys := Matrix([[diff(x(t), t), diff(y(t), t)], [diff(y(t), t), -(diff(x(t), t))]]) = Matrix([[(-x(t)*t+y(t))*x(t)+(x(t)+y(t)*t)*y(t)+2*x(t)*c, (-x(t)*t+y(t))*y(t)-(x(t)+y(t)*t)*x(t)+2*y(t)*c], [(-y(t)*t-x(t))*x(t)+(-x(t)*t+y(t))*y(t)+2*y(t)*c, (-y(t)*t-x(t))*y(t)-(-x(t)*t+y(t))*x(t)-2*x(t)*c]])](/support/helpjp/helpview.aspx?si=8699/file01351/math177.png)
| (5) |
The two coupled odes are
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![ode[1] := lhs(sys)[1, 1] = rhs(sys)[1, 1]](/support/helpjp/helpview.aspx?si=8699/file01351/math183.png)
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![ode[1] := diff(x(t), t) = (-x(t)*t+y(t))*x(t)+(x(t)+y(t)*t)*y(t)+2*x(t)*c](/support/helpjp/helpview.aspx?si=8699/file01351/math186.png)
| (6) |
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![ode[2] := lhs(sys)[1, 2] = rhs(sys)[1, 2]](/support/helpjp/helpview.aspx?si=8699/file01351/math190.png)
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![ode[2] := diff(y(t), t) = (-x(t)*t+y(t))*y(t)-(x(t)+y(t)*t)*x(t)+2*y(t)*c](/support/helpjp/helpview.aspx?si=8699/file01351/math193.png)
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The matrix solution to this matrix system of equations with initial conditions  at ![t[0] = 0](/support/helpjp/helpview.aspx?si=8699/file01351/math198.png) is computed as:
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Recalling the form of  , the solution to the system of odes is constructed from  as
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![sol := {x(t) = matrix_sol[1, 1], y(t) = matrix_sol[1, 2]}](/support/helpjp/helpview.aspx?si=8699/file01351/math213.png)
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This result can be verified with odetest
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![odetest(sol, [ode[1], ode[2]])](/support/helpjp/helpview.aspx?si=8699/file01351/math225.png)
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![[0, 0]](/support/helpjp/helpview.aspx?si=8699/file01351/math228.png)
| (9) |
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