Systems of ODEs

Brown University, Applied Mathematics

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Systems of ODEs

The tutorial accompanies the textbook Applied Differential Equations. The Primary Course by Vladimir Dobrushkin, CRC Press, 2015; http://www.crcpress.com/product/isbn/9781439851043

We consider a system of nonlinear differential equations in normal form (when the derivatives are isolated):

\[ \begin{cases} {\text d}x_1 /{\text d}t &= g_1 (t, x_1 , x_2 , \ldots , x_n ), \\ {\text d}x_2 /{\text d}t &= g_2 (t, x_1 , x_2 , \ldots , x_n ), \\ \vdots & \qquad \vdots \\ {\text d}x_n /{\text d}t &= g_n (t, x_1 , x_2 , \ldots , x_n ), \end{cases} \]
where \( g_k (t, x_1 , x_2 , \ldots , x_n ) , \quad k=1,2,\ldots , n, \quad \) is a given function of \( n+1 \) variables. INstead of \( {\text d}x /{\text d}t \) we will use either of shorter notations \( x' \) or the more customary notation \( \dot{x} \) to denote a derivative of \( x(t) \) with respect to variable t associated with time. If the slope functions do \( g_k (t, x_1 , x_2 , \ldots , x_n ) , \quad k=1,2,\ldots , n, \quad \) do not depend on time variable t, the system of differential equations is referred to as autonomous:

\[ \begin{cases} {\text d}x_1 /{\text d}t &= g_1 (x_1 , x_2 , \ldots , x_n ), \\ {\text d}x_2 /{\text d}t &= g_2 (x_1 , x_2 , \ldots , x_n ), \\ \vdots & \qquad \vdots \\ {\text d}x_n /{\text d}t &= g_n (x_1 , x_2 , \ldots , x_n ), \end{cases} \]
Differential equations are usually sugject to the initial conditions:
\[ x_1 (t_0 ) = x_{10} , \quad x_2 (t_0 ) = x_{20} , \quad \ldots , \quad x_n (t_0 ) = x_{n0} , \]

where \( t_0 \) is a specified value of t and \( x_{10} , x_{20} , \ldots , x_{n0} \) are prerscribed constants. The problem of finding a solution to a system of differential equations that satisfies the givne initial conditions is called an initial value problem.

When the functions \( g_k (t, x_1 , x_2 , \ldots , x_n ) , \quad k=1,2,\ldots , n, \quad \) are linear functions with respect to n dependent variables \( x_1 (t), x_2 (t), \ldots , x_n (t), \) we obtain the general system of first order linear differential equations in normal form:

\[ \begin{cases} \dot{x}_1 &= p_{11} \,x_1 + p_{12} (t)\, x_2 + \cdots + p_{1n} (t)\, x_n + f_1 (t), \\ \dot{x}_2 &= p_{21} \,x_1 + p_{22} (t)\, x_2 + \cdots + p_{2n} (t)\, x_n +f_2 (t), \\ \vdots & \qquad \vdots \\ \dot{x}_n &= p_{n1} \,x_1 + p_{n2} (t)\, x_2 + \cdots + p_{nn} (t)\, x_n +f_n (t). \end{cases} \]

In this system of differential equations, the \( n^2 \) coefficients \( p_{11}, p_{12}, \ldots , p_{nn} \) and the n functions \( f_1 (t) , f_2 (t) , \ldots , f_n (t) \) are assumed to be known. If the coefficients \( p_{ij} \) are constants, we have a constant coefficient system of equations. Otherwise, we have a linear system of differential equations with variable coefficients. The system id said to homogeneous or undriven if\( f_1 (t) \equiv f_2 (t) \equiv \cdots f_n (t) \equiv 0. \)

The linear system of differential equations can be written in compact vector form:

\[ \dot{\bf x} = {\bf P} (t)\, {\bf x} + {\bf f}(t) , \]
where \( {\bf P} (t) \) denotes the following square matrix:
\[ {\bf P} (t) = \begin{bmatrix} p_{11} (t) & p_{12} (t) & \cdots & p_{1n} (t) \\ p_{21} (t) & p_{22} (t) & \cdots & p_{2n} (t) \\ \vdots & \vdots & \vdots & \vdots \\ p_{n1} (t) & p_{n2} (t) & \cdots & p_{nn} (t) \\ \end{bmatrix} \]
Here \( {\bf x} (t) \) and \( {\bf f} (t) \) are n-dimensional vector-functions that are assumed to be columns.

In MuPad, defining matrices is done similar to Maple:

A1 := matrix([[3,2,4],[2,0,2],[4,2,3]])

phi1 := exp(A1*t)

phi1_dot := diff(phi1,t)

testeq(phi1_dot,A1*phi1)

A2 := matrix([[1,-1,-2],[1,3,2],[1,-1,2]])

phi2 := exp(A2*t)

phi2_dot := diff(phi2,t)

testeq(phi2_dot,A2*phi2)

reset

A3 := matrix([[-15,-7,4],[34,16,-11],[17,7,5]])

phi3 := exp(A3*t)

phi3_dot := diff(phi3,t)

testeq(phi3_dot,A3*phi3)

reset
a1 := x'(t) = -.1/20*x(t)

a2 := y'(t) = -x'(t) - .1/40*y(t)

a3 := z'(t) = .1/40*y(t) - .1/50*z(t)

fun := {a1,a2,a3,x(0)=15,y(0)=0,z(0)=0}

sol := ode::solve(fun,{x(t),y(t),z(t)})

x1(t) := sol[1][2]

x2(t) := sol[1][1]

x3(t) := sol[1][3]

% distinct eigenvalues
A := matrix([[5,9],[6,2]])
x := (t^L)*z
x_dot := diff(x,t)
one := matrix([[L,0],[0,L]])
y := det(one-A)
factor(y)
linalg::eigenvectors(A)


 



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