M427K Handout: Second Order Homogeneous ODEs withConstant CoefficientsSalman ButtFebruary 2, 2006This handout discusses second order ordinary differential equations, specifically homogeneousequations with constant coefficients.PreliminariesRecall the the form of a general second order differential equation:y00=d2ydt2= f(t, y,dydt) = f(t, y, y0) (1)We say (1) is linear if the function f has the formIn this case (i.e. if (1) is linear), we write our equation asFor second order ODEs, we will need two initial values if we are to determine the particularsolution from the general solution (i.e. determine the constants):y(t0) = y0, y0(t0) = y00Our second order ODE is said to be homogeneous if g(t) is zero for all t; otherwise the equa-tion is nonhomogeneous. We say that a second order homogeneous ODE has constant coefficientsifSecond Order Homogeneous ODEs with Constant CoefficientsWe write our differential equation asay00+ by0+ cy = 0Using some inspiration from when we studied the equation y00− y = 0, we guess that a s olutionto this differential equation may be ert. Let’s try our guess out: substituting y = ertinto ourequation, we have1Thus we have what is called the characteristic equation associated to the differential equa-tion:We find r by using the quadratic formula:r =For now, we will assume the roots are real and distinct and denote them by r1and r2. Thisgives us two solutions: y1(t) = er1t, y2(t) = er2t. Observe that indeed a linear combination of thesetwo solutions is also a solution:Thus we have our general solution for a second order homogeneous ODE with constant coeffi-cients (with real, distinct roots of its associated characteristic equation). To determine c1, c2, wemerely use our initial conditions to get a two dimensional linear system which we solve to find ourconstants. Let’s see some examples: consider the differential equationsy00− y0− 2y = 0, 2y00− 3y0− 2y = 0, 6y00+ 5y0− 6y =
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