M427K Handout: First Order ODEsSalman ButtJanuary 19, 2006This handout discusses the theory of first order (ordinary) differential equations. That is, wewill be looking for solutions to differential equations of the formdydt= f(t, y), (1)which may also be writteny0= f(t, y). (2)Observe that t is the only independent variable and y = y(t) is a function of t. To say this differentialequation is first order is to say that the highest derivative appearing is the first derivative. We willalso assume for now that the equation is linear – i.e. the function f depends linearly on y. Thusour differential equation takes on the formy0= −py + g (3)where p, g are functions of t (note that the minus sign in front of the p is just smart bookkeeping,as you will see). Now what does it mean to find a solution to this (or any) differential equation?Note that a solution must satisfy its own differential equation – this is a good way to check yoursolution. There are two basic types of solutions to a differential equation (first order or otherwise):the general solution is a solution that does not use any initial data. Formally, it is a familyof solutions depending on sufficiently many parameters to give all but finitely many solutions. Aparticular solution is a solution where the initial data is used to compute constants in the generalsolution to yield one solution. There are a few ways to arrive at these solutions, so let us look atone method: ˙Method of Integrating FactorsWe begin with a general first order linear differential equation, which we write asy0+ p(t)y = g(t). (4)What we are looking for is a function µ(t) called the integrating factor which has the very specialproperty that(5)If we find such a function, we then see1Now from Equation (5) we haveµ0y + µy0= (µy)0= µ(y0+ py) = µy0+ µpy. (6)This then tells us thatµ0y = µpy =⇒ µ0= µp. (7)Rewriting this equation, we getµ0µ= p, (8)which we integrate on both sides with respect to t to get(9)We choose our constant C = 0 to make life simple and exponentiate both sides to get our finalform for the integrating factor(10)With our integrating factor in hand, we can now solve our general first order linear differentialequation:y(t) = . (11)Example: Consider the differential equationy0+2xy =cos xx2(12)Our integrating factor is merelyAnd so our solution is readily seen to