Advanced Engineering Mathematics
5 Linear Systems of ODEs 5.1 Systems of ODEs In a sense, Chapter 5 equals Chapter 2 “plus” Chapter 3, in the sense that Chapter 5 combines use of matrix theory and ordinary differential equation (ODE) methods. When we have more than one linear ODE, results from matrix theory turn out to be useful. Example 5.1 For the circuit shown in Figure 5.1, let v(t) be the voltage drop across the capacitor and I(t) be the loop current. The input V(t) is a given function. Assume, as usual, that L, R, and C are constants. Write down a system of ODEs in R2 that models this circuit. Method: The series RLC circuit shown in Figure 5.1 is analogous to the DC series RLC circuit discussed near the end of Section 3.3. The first ODE models the voltage drop across the capacitor being v(t) = 1 q(t),whereq(t) is the charge on the capacitor and C ˙ q˙(t) = I(t). The second ODE in the system is Kirchhoff’s voltage law, LI(t)+RI(t)+v(t) = V(t), after dividing through by L. The system is ⎧ ⎫ ⎨˙( ) = 1 ( ) ⎬ v t C I t ⎩ ⎭ . (5.1) ˙( ) = 1 ( ( ) − ( ) − ( )) I t L V t RI t v t More generally, consider a system of two ODEs in unknowns x1(t), x2(t): ⎧ ⎫ ˙ ⎨x1(t) = F1 t, x1(t), x2(t) ⎬ ⎩ ⎭ . (5.2) ˙ x2(t) = F2 t, x1(t), x2(t) A special case is ⎧ ⎫ ˙ ⎨x1(t) = a11(t)x1 + a12(t)x2 + f1(t)⎬ ⎩ ⎭ , (5.3) ˙ x2(t) = a21(t)x1 + a22(t)x2 + f2(t) which is called a linear system.
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