REVIEW NOTES ON DIFFERENTIAL EQUATIONS PETE L CLARK AND ROBIN GOTTLIEB 1 Basic concepts for differential equations A differential equation is an equating relating an unknown function to its derivative s It is easiest to give examples in all of the following y is a function of t 1 dy dt 1 1 t2 2 dy dt 3y 3 dy dt t y 4 y 00 3y 0 2y 0 The order of a differential equation is the highest numbered derivative which appears in the equation Equations 1 through 4 are first order differential equations equation 5 is a second order differential equation By the way one could certainly consider differential equations of higher order e g the equation y 10 0 is a tenth order differential equation which asks for all functions whose tenth derivative is identically zero Any degree nine polynomial a9 t9 a8 t8 a1 t a0 is a solution to this equation But in our course we have had our hands full with first order equations and very particular second order equations 1 To say that a function y y t is a solution to a differential equation just means that if we subtitute it into both sides of the equation we do indeed get an equality For example the function y1 t arctan t is a solution to equation 1 because the derivative 1 of arctan t is 1 t 2 It is not the only solution y t arctan t C also works Existence and uniqueness of solutions first order case The general firstorder differential equation is dy dt f t y where f is just any function of t and y Notice that equations 1 through 4 are of this form for 4 bring the y 2 to the other side A first order differential equation never has just one solution but rather an one parameter family of solutions We specify a particular solution by 1Notice that all our differential equations above are in the form y 0 some function of y and t or y 00 some function of y 0 y and t in fact as part of our definition of a differential equation which wasn t very precise we should understand that we mean a differential equation of this form or we could have problems For example Show that there is no function y y t such that dy dt 2 1 t2 y 2 Hint look at the signs of the left hand side and the right hand side of the equation Solution The left hand side is strictly positive and the right hand side is less than or equal to zero No way 1 2 PETE L CLARK AND ROBIN GOTTLIEB requiring the solution to take a certain value at a certain point notice that this is a generalization of the situation of taking antiderivatives in the differential equation dy dt f t the general solution is y F t C and we can pick out a particular choice of antiderivative by imposing a condition y t0 y0 or y0 F t0 C so that C y0 F t0 This leads to the following terminology Initial value problem A first order initial value problem is a first order differential equation together with an initial condition dy dt f t y y t0 y0 Now we have the theoretical result that we have exploited so many times in our study of differential equations Theorem 1 Existence and uniqueness theorem for first order equations Any first order differential equation of the form dy dt f t y with initial condition y t0 y0 where f t y is well behaved 2 has a unique solution y i e there is a unique function y satisying the differential equation and such that y t0 y0 We can also view the collection of all solutions to a given first order equation dy dt f t y geometrically we view each one as a solution curve in the t y plane We can reinterpret the existence and uniqueness theorem as telling us that i every point on the t y plane lies on a solution curve and ii no point lies on more than one solution curve Equivalently we say that the solution curves fill up the plane and never cross Initial value problems for second order differential equations To specify a particular solution to a second order differential equation will require two initial conditions Again this can be predicted from the differential equation y 00 f t which asks for all functions F whose second derivative is f We can solve this equation in two steps y 0 F t C1 where F 0 f and y F t C1 t C2 where F 0 F Thus we have a modified Initial value problem A second order initial value problem is a second order differential equation together with two initial conditions y 00 f t y y 0 y t0 y0 y 0 t0 v0 And just as for first order equations we have the Theorem 2 Existence and uniqueness theorem for second order equations Any initial value problem y 00 f t y y 00 y t0 y0 y 0 t0 v0 has a unique solution provided f is well behaveds 2 Taxonomy of first order differential equations We have studied several different kinds of first order differential equations 2by well behaved we mean that they will have infinitely many derivatives when regarded as either functions of t or y All the functions we deal with in this course will be well behaved REVIEW NOTES ON DIFFERENTIAL EQUATIONS 3 2 1 Integration equations An equation of the form dy dt f t can be thought of as an integration equation because it simply asks for an indefinite integral of f t I e if F 0 f is any antiderivative the general solution is y F t C note well that the solution curves are vertical translates of each other Since solving integration equations requires no new techniques we won t see many of them but they are good to keep in mind as well understood special cases to understand how the theory should go e g from them we could guess the need for initial conditions Geometrically an integration equation says that the slope depends only upon t therefore the solutions are vertical translates 2 2 Autonomous equations An equation of the form dy dt f y is called an autonomous equation Compare with the previous example the function on the right hand side is only a function of y and hence independent of time autonomous What does the time independence of solutions mean It means that all solution curves passing through the same y value y y0 are governed by the same law dy dt f y0 Thus if we take any solution curve y t and push it to the left or the right what the differential equation is telling it to do at any particular y value is not changing and we find that solution curves to an autonomous equation are invariant under horizontal translation 1 Autonomous equations lend themselves well to a qualitative analysis Procedure for …