Math0 Lecture # Introduction to Differential Equations Basic definitions Definition : (What is a DE?) A differential equation (DE) is an equation that involves some of the derivatives (or differentials) of an unknown function with respect to one or more variables. Definition : (Independent variable) It is the variable that derivative is taken with respect to it. Definition : (Dependent variable) It is the variable where its derivative occurs in the DE. Definition : (Ordinar DE) It is a DE that has onl one independent variable. Definition 5: (Partial differential equation) It is a DE that have more than one independent variable. Definition 6: (Order of the DE) It is the value of the highest order derivative appearing in the DE. Definition 7: (Degree of the DE) It is the eponent of the highest order derivative appearing in the DE. Definition8: (Linearit of the DE) When the dependent variable as well as all of its derivatives appearing in the DE are of power one and no multiplication combinations among them, we call the DE a linear DE. Otherwise, we call it nonlinear DE. Page
Math0 Lecture # Eample : (i) Determine the order and the degree of the given differential equations, also state whether it is linear or nonlinear: (ii) t d dt d dt d t sint dt d dt t (iii) e (iv) sin t sin t (v) d d t dt dt cos t t (vi) sin e Solution: (i) nd order, st degree and linear. (ii) st order, st degree and nonlinear. (iii) th order, nd degree and nonlinear. (iv) rd order, st degree and nonlinear. (v) st order, st degree and linear. Definition 9: (Solution of the ODE) A function () is said to be an eplicit solution of the n th. Order ODE if () and all of its derivatives up to order n eist in some interval I and satisf the given DE in I. In some cases we are not able to epress as an eplicit function of, in such cases we sa that we have an implicit solution of the DE. Page
Math0 Lecture # Tpes of solutions of ODE's An n-th. order ODE ma have three kinds of solutions: () General solution It is a solution that contain n arbitrar constants. () Particular solution It is a solution that can be deduced from the general solution b assigning certain numerical values to the arbitrar constants. () Singular solution It is a solution of the D. E. that can not be deduced from an general solution. Page
Math0 Lecture # Eample : Verif that the given function is a particular solution for the given differential equation 0t sec t, cost cost tsint, Solution: cost cost tsint sint cost tcost sin t cost cost cost sin t cost t sin t From : sin t cost cost cost tsint cost sin t cost cost cos cost t cost sect cost cost sect Page
Math0 Lecture # First Order Differential Equations Standard Forms of First Order Differential Equations: F ;, 0 () = f, () M, d N, d 0 () Eamples 0 () () d d 0 () - Separable DE If the first order DE can be rearranged in the form A()d = B()d then it is called a separable DE. Its solution can be obtained b direct integration as follows: A( )d B( )d Eample Solve the following DE :, 0 d d ( )d d d d 0 d d c c ec k Solution k 5 Page
Math0 Lecture # Initial value problems (IVP) In order to determine a specific value to the arbitrar constant appearing in the general solution we must have one additional condition called the initial condition (IC). In this case the DE together with the initial condition are called an initial value problem (IVP). Eample 5: Solve the initial value problem ( ) d d 0 (DE), (0) = (IC) Solution: the general solution of the DE is d d 0 tan tan Appling the initial conditions we get: tan 0 tan c 0 c c Therefore, the solution of the IVP is: tan tan c. 6 Page
Math0 Lecture # Solved Problems Find the general solution for each of the following differential equations: (i) B separation d d d d solution) C (Eplicit (ii) e e e B separation e d e d e e d e e d e d e e d e e e e C (Implicit solution) (iii) B separation d d 7 Page
Math0 Lecture # 8 Page d d d d C 9 (Implicit solution) (iv) 0 cos sin d d B separation sec tan cos sin cos sin d d d d d ()() tan C (Implicit solution) (v) 8 B separation
Math0 Lecture # 9 Page d d d d d d d d d d 5 5 5 5 C 5 5 (Implicit solution) Good luck Dr. Osama Shahin