Lecture 8

1. Lecture 8 Second Order Linear Differential Equations Lecture 8

2. Second Order Differential Equations A second order differential equation is an equation involving the unknown function y, its derivatives y' and y'', and the variable x: We will only consider explicit differential equations of the form, Lecture 8

3. Arbitrary constants of Second Order Equations Since a second order equation involves a second derivative and hence, two integrations are required to find a solution, the solution usually contains two arbitrary constants. Lecture 8

4. Equations with missing Nonlinear second order differential equations are difficult to solve for general f: but there are two cases where simplifications are possible: • Equations with the y missing Let v = y'. Then the new equation satisfied by v is This is a first order differential equation. Once v is found its integration gives the function y. Lecture 8

5. Equations with missing (2) Equations with the x missing Let v = y'. Since We get This is again a first order differential equation. Once v is found then we can get y through Lecture 8

6. The Linear Case A linear second order differential equations is written as associate the so called associated homogeneous equation For the study of these equations we consider the explicit ones given by Lecture 8

7. Solution of the Linear Case where p(x) = b(x)/a(x), q(x) = c(x)/a(x) and g(x) = d(x)/a(x). If p(x), q(x) and g(x) are defined and continuous on the interval I, then the IVP has a unique solution defined on I. The general solution to the equation (NH) is given by where is the general solution to the homogeneous associated equation (H); is a particular solution to the equation (NH). Lecture 8

8. Solution of the Linear Case 2 In conclusion, we deduce that in order to solve the nonhomogeneous equation (NH), we need to : Step 1: find the general solution to the homogeneous associated equation (H), say Step 2: find a particular solution to the equation (NH), say Step 3: write down the general solution to (NH) as Lecture 8

9. Homogeneous Linear Equations Consider the homogeneous second order linear equation or the explicit one Basic property:If and are two solutions, then is also a solution for any arbitrary constants and Lecture 8

10. Linear Independence of Solutions Let y1 and y2 be two differentiable functions. The Wronskian W(y1,y2), associated to y1 and y2, is the function Lecture 8

11. Properties of the Wronskian (1) If y1 and y2 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then (2) If y1 and y2 are two solutions of the equation y'' + p(x)y' + q(x)y = 0, then In this case, we say that y1 and y2 are linearly independent. (3) If y1 and y2 are two linearly independent solutions of the equation y'' + p(x)y' + q(x)y = 0, then any solution y is given by y=c1y1+c2y2 for some constant c1 and c2. In this case, the set {y1,y2}is called the fundamental set of solutions. Lecture 8

12. Homogeneous Linear Equations with Constant Coefficients A second order homogeneous equation with constant coefficients is written as where a, b and c are constant. This type of equation is very useful in many applied problems (physics, electrical engineering, etc..). Let us summarize the steps to follow in order to find the general solution: (1) Substitute . Then we can write down the characteristic equation This is a quadratic equation. Let r1 and r2 be its roots. If r1 and r2 are distinct real numbers (if D>0), then the general solution is Lecture 8

13. Equations with Constant Coefficients 2 (3) If r1=r2 (which happens if D=0 ), then the general solution is (4) If r1 and r2 are complex numbers (which happens if D<0), then the general solution is where Lecture 8