Differential Equations Chapters 1‐3
§1.1: Definitions and Terminology Differential Equation: An Equation containing the derivatives of one or more independent variables with respect to(wrt) one or more independent variables
Classifications By Type: o Ordinary Differential Equations (ODE) (ODE):: An equation that contains only ordinary derivatives of one or more dependent depende nt variables wrt a single independent variable
o
ex: 0
Classification by order: o The order of a differential equation is the order of the highest derivative in the equation.
5 , 6 0, 2 2
Partial Differential Equations (ODE) (ODE):: An equation that contains partial derivatives of one or more dependent variables wrt two or more independent variables
ex:
ex:
→ 0, → 1, → 2
Classification by linearity: th o An n order differential equation is linear if the following conditions are true: ’ ’’ st The dependent variable “y” & it’s derivatives (y ,y ) are of the 1 degree ’ ’’ The coefficients of a0, a1,a2… Of y, y ,y … depend on at most x (they will not be depend on y or any other variable) Linear Equation takes form:
..
Solution: Any function ϕ, defined on an interval I and possessing at least n derivatives that are continuous on I, which when substituted substituted into an nth-order ODE ODE reduces the equation to an identity, is said to be a solution of the equation on the interval.
Interval of existence: The domain of a solution o Explicit vs. Implicit solutions An Explicit Solution is a solution where the dependent variable is expressed solely in terms of the independent variable is explicit A relation G(x, y) = 0 is said to be an Implicit Solution of an ordinary differential equation (4) on an interval I, provided that there exists at least one function that satisfies the relation as well as the differential equation on I.
§1.2: Initial Value Problems
Differential Equations Chapters 1‐3
An Initial Value Problem(IVP) is an ODE that is paired with conditions imposed on unknown function y(x)
First order IVP o
o
, subject to Solve
Second order IVP
,,′ subject to and Existence and Uniqueness Existence: does the ODE , possess any solutions? Di any of them pas through , o
Solve
o
o
o
Uniqueness: When Can we be certain that there is precisely one solution passing through
o
Theorem: Existence of a unique solution Let R be a rectangular region in the xy-plane defined by a
o
,
xb;cyd
thatcontains,initsinterior.If, arecontinuousonR
thenthereexistsomeintervalI0containedina,b,andauniquefunction yxdefinedonI0thatisasolutiontotheivp Inotherwords,UniqueSolutions ExistWhere ,is continuous
is contiuous
§2.2: Separable Variables A first-order differential equation of the form separable variables.
is said to be separable or to have
To Solve a Separable variable equation: 1) Group all y- terms on one side of the equation, all x-terms on the other side. dy & dx must both be in the numerator 2) 3)
Integratebothsides Ifanexplicitsolutionisneeded;solvefory →
Differential Equations Chapters 1‐3
Ex: Quiz 2:
2 2 3 3 §2.3: Linear Equations A first order DFEq in the form variable y
is a linear equation for the dependent
If g(x)= 0 then the function is homogeneous
If g(x) o
o
0 then the function is non-homogeneous Standard form found by dividing above equation by giving: P where P
To Solve First Order ODE by integrating factor (I.F):
1) Find the standard form by deviding by the coefficient of dy/dx 2) Use P(x) from standard form to identify the integrating factor for the ODE
.
3) Multiply all terms of standard form by I.F. Left hand side always reduces to the derivative of your integrating times y so the result is
∙ ∙
4) Integrate both sides and solve for y
Any term whose contribution to the solution becomes negligible( approaches zero) as x is called a transient term
→∞
Any point which makes point of the ODE
0
causes a discontinuity on P(x) is called a singular
§2.4: Exact Differential and Exact Equation
,, Mx,ydxNx,ydy 0
A differential expression is an exact differential in a region R of the xy-plane if it corresponds to the differential of some function f(x,y) defined in R. A first order ODE of the form is an exact equation if the Left Hand Side is an exact differential.
al Equations Chapters 1‐3
riterion For an Exact Differential E uation o The necessary an sufficient ondition th t differ ential is: o
Mx,yd Nx, y y be an ex ct
olving an E act Equation
-
1) Integ ate M with espect to x. Keep in mi d, the arbitrary consta t maybe a function of y[(g ( )]. This yields function f(x ,y). 2) Diffe entiate the r esult f(x ,y). with respe t to y ( 3) Set
Nx,y
).
nd solve fo "arbitrary onstant"
4) Set s lution f(x, y) equal to a arbitrary c onstant C
Alter atively you can work ith N 1) Integ ate N with r espect to y. Keep in mi d, the arbit ary constant maybe a function of x[(g ( )]. This yields function f(x ,y). -
2) Diffe entiate the r esult f(x ,y). with respe t to x ( 3) Set
Mx,y
).
nd solve fo "arbitrary onstant"
4) Set s lution f(x, y) equal to a arbitrary c onstant C tegrating f ctors o It is possible to fi d an integr ting factor, which will transform a non-exact equat on into an exact equati n.
If i If i
a function of x alone, integrating f ctor is a function of y alone, integrating f ctor is
Differential Equations Chapters 1‐3
§2.5: Solutions by Substitution HOMOGENEOUS EQUATIONS a function f that possesses the property ) for some real number α , then f is said to be a homogeneous function of degree α
, ,
Mx,ydxNx,ydy 0issaidtobehomogeneous*ifbothcoefficientfunctions MandNarehomogeneousequationsofthesamedegree
To solve homogeneous equations: 1 Substitute 2 Simplify separable equation 3 Solve using separation of variables
or →
Ex: HW set 3
yx; Nx,y x Mx,y y , ;,
‐Botharehomogeneousequationswithα2 , ‐Substituteyanddyforeq. above 0 0 2 0 2 1 2 1 2 1 12 12 2 ln|| ln|| ln| 2| ln|| ln ln 2 ln|| ln 2
Quick Review: Partial Fraction Decomposition
1 2 2 Let u=0; 1 2 1 2 1 2, 12 Let u=-2 1 2, 12
BERNOULI EQUATIONS
P Let Find and substitute o
o
REDUCTION TO SEPERABLE VARIABLES
can be reduced by letting where B0 find and substitute
