REVIEW NOTES ON DIFFERENTIAL EQUATIONS PETE L CLARK FOR THE MOST PART 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 y0 y2 1 t 5 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 Notice 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 Problem 1 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 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 1 the derivative 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 first order 1 2 PETE L CLARK FOR THE MOST PART 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 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 initial value problem dy dt f t y y t0 y0 where f t y is well behaved 1 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 solution curves foliate the plane which is fancy terminology for saying that they fill up the plane and never cross A warning about Theorem 1 The existence part of this theorem should be interpreted as saying that through any point given any initial condition is some solution curve It is NOT true that any solution curve can be continued indefinitely as a function of t solution curves may not be defined for all time Example 1 Consider the differential equation dy dt y 2 The general solution is 1 y t C t each of these solutions has a vertical asymptote at t C e g the 1 solution t is undefined at t 0 you should sketch a picture of this function In particular y t 1 t passes through the point 1 1 and goes to as we approach t 0 through negative values Then for positive values of t the solution we have written down switches to the other branch of the hyperbola but a solution curve must represent an unbroken path in the plane so we regard the piece of 1 t for positive t as representing a different solution to the differential equation Notice that in any case it is NOT correct to say If y t0 0 then limt y t because any solution which is positive at some value of t will go to infinity in finite time and not be positive for very large values of t We will return to this point when we talk about autonomous equations 1by 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 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 That is if F is a second antiderivative of f we can add any linear function and get another second antiderivative and there is a two parameter family of linear functions 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 The same caveat about solutions not necessarily existing for all values of time applies although as we shall see later this problem does not arise for linear differential equations …