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Unformatted text preview: 2 MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 More Definitions. Consider a linear second order differential equation a(t) y ￿￿ + b(t) y ￿ + c(t) y = f (t). If we have initial conditions in the form dy ￿ (t0 ) = y0 ; dt we call this system an initial value problem. Note that an initial value problem specifies (1) an initial point (t0 , y0 ), and ￿ (2) an initial slope y0 at that point. We say that the differential equation is homogeneous if f (t) = 0 for all t i.e., if the differential equation is in the form a(t) y ￿￿ + b(t) y ￿ + c(t) y = 0. y (t0 ) = y0 and The functions a(t), b(t), and c(t) are called the coefficients of the differential equation. Any second order differential equation which cannot be placed in this form is said to be nonhomogeneous equation or an inhomogeneous equation. Constant Coefficient Equations. Say for the moment that the coefficients of the differential equation are constant functions: a(t) = a, b(t) = b, and c(t) = c. We consider equations in the form a y ￿￿ + b y ￿ + c y = f (t). Say for the moment that this is a homogeneous equation. We give a trick to solve such an equation. For inspiration, consider the analogue of a constant coefficient homogeneous equation of the first order: a y ￿ + b y = 0. We can solve this equation because it is separable: a y ￿ = −b y 1 dy b =− y dt a d b ln |y | = − dt a b ln |y | = − t + (constant) =⇒ y (t) = (constant) · ert a in terms of the constant b r=− =⇒ a r + b = 0. a Hence the solution to the differential equation a y ￿ + b y = 0 is essentially the exponential y = ert for some constant r. Note that the equation a r + b = 0 is similar in form to the differential equation. Now we return to the constant coefficient homogeneous equation of the second order. We will guess that the solution is y = ert for some constant r. We have the derivatives y (t) = ert , y ￿ (t) = r ert , and y ￿￿ (t) = r2 ert . This gives the relation ￿ ￿ ￿ ￿ ￿￿￿ ￿ 0 = a y ￿￿ + b y ￿ + c y = a r2 ert + b r ert + c ert = a r2 + b r + c ert . The exponential ert ￿= 0, so we must have a r2 + b r + c = 0. This equation is called the characteristic equation of the homogeneous differential equation. We will show the following. Consider the constant coefficient homogeneous equation a y ￿￿ + b y ￿ + c = 0. ...
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