Systems

# Between these solutions we finally find the polar

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between these solutions, we finally find the polar equation of the orbits: r = r 0 e a ( θ θ 0 ) t/b . If you graph this for a negationslash = 0 , you will get stable or unstable spirals. Example 2.12. Consider the specific system x = y + x y = x + y. (2.31) In order to convert this system into polar form, we compute rr = xx + yy = x ( y + x ) + y ( x + y ) = r 2 . r 2 θ = xy yx = x ( x + y ) y ( y + x ) = r 2 . This leads to simpler system r = r θ = 1 . (2.32) Solving these equations yields r ( t ) = r 0 e t , θ ( t ) = t + θ 0 . Eliminating t from this solution gives the orbits in the phase plane, r ( θ ) = r 0 e θ θ 0 .

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2.3 Matrix Formulation 43 A more complicated example arises for a nonlinear system of differential equations. Consider the following example. Example 2.13. x = y + x (1 x 2 y 2 ) y = x + y (1 x 2 y 2 ) . (2.33) Transforming to polar coordinates, one can show that In order to convert this system into polar form, we compute r = r (1 r 2 ) , θ = 1 . This uncoupled system can be solved and such nonlinear systems will be studied in the next chapter. 2.3 Matrix Formulation We have investigated several linear systems in the plane and in the next chapter we will use some of these ideas to investigate nonlinear systems. We need a deeper insight into the solutions of planar systems. So, in this section we will recast the first order linear systems into matrix form. This will lead to a better understanding of first order systems and allow for extensions to higher dimensions and the solution of nonhomogeneous equations later in this chapter. We start with the usual homogeneous system in Equation (2.5). Let the unknowns be represented by the vector x ( t ) = parenleftbigg x ( t ) y ( t ) parenrightbigg . Then we have that x = parenleftbigg x y parenrightbigg = parenleftbigg ax + by cx + dy parenrightbigg = parenleftbigg ab cd parenrightbiggparenleftbigg x y parenrightbigg A x . Here we have introduced the coefficient matrix A . This is a first order vector differential equation, x = A x . Formerly, we can write the solution as x = x 0 e At . 1 1 The exponential of a matrix is defined using the Maclaurin series expansion
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