D.E. Definitions

  1. Differential Equation
    An equation containing derivatives or differentials with one or more dependent variables
  2. Ordinary differential equation
    An ordinary d.e. contains only ordinary derivatives or differentials of one or more dependent variables with respect to a single independent variable
  3. Partial differential equation
    An equation involving partial derivatives of one or more dependent variable(s) with respect to two or more independent variables
  4. Order of a differential equation
    the order of a d.e. is the highest order derivative in the d.e.
  5. Linear or Nonlinear differential equations
    • a d.e. or order n is said to be linear if it has the form:
    • chart?chf=bg,s,00000000&cht=tx&chl=A_n(x)%5Cfrac%7Bd%5Eny%7D%7Bdx%5En%7D%2BA_%7Bn-1%7D(x)%5Cfrac%7Bd%5E%7Bn-1%7Dy%7D%20%20%7Bdx%5E%7Bn-1%7D%7D%2B&chs=426x78...chart?chf=bg,s,00000000&cht=tx&chl=%2BA_%7B2%7D(x)%5Cfrac%7Bd%5E%7B2%7Dy%7D%7Bdx%5E%7B2%7D%7D%2BA_%7B1%7D%20%20(x)%5Cfrac%7Bd%5E%7B%7Dy%7D%7Bdx%5E%7B%7D%7D&chs=340x78chart?chf=bg,s,00000000&cht=tx&chl=%2BA_%7B0%7D(x)y%3Dg(x)&chs=244x38

    Image Upload 1otherwise it is called an nonlinear d.e.
  6. In a linear d.e.,
    • a) The dependent variable y and its derivatives y,y',y'',...,yn are of the first degree, that is, the order of each is one.
    • b) The coefficients of y,y',y'',...,yn depend at most on independent variable x
  7. A solution of a differential equation
    is a function f defined on some interval I, which when substituted into the d.e. reduces the equation to an identity
  8. If a function is _____ at a, it must be ______ at a
    If a function is differentiable at a, it must be continuous at a.

    But the converse is NOT TRUE.
  9. Part I of the Fundamental Theory of Calculus
    If f(t) is a continuous function on [a,b], define chart?chf=bg,s,00000000&cht=tx&chl=G(x)%3D%5Cint_a%5Exf(t)dt&chs=248x56, then chart?chf=bg,s,00000000&cht=tx&chl=G'(x)%3D%5Cfrac%7Bd%7D%7Bdx%7D%5B%5Cint_a%5Exf(t)dt%5D%3Df(x)&chs=440x68
  10. Implicit function
    Given an equation relating x and y or a relation G(x,y)=0, if we can solve y in terms of x, then y=f(x) is called an implicit function defined by the given equation, that satisfies the equation.
  11. An implicit solution of a d.e.
    A relation G(x,y)=0 is an implicit solution to the given d.e. if there exists an implicit function y=f(x) defined on some interval I which satisfies the relation G(x,y)=0 as well as the d.e. on I.
  12. Particular solution to a d.e.
    A solution to the given d.e. is called a particular solution if it does not contain any parameters in it.
  13. Higher Order Partial Derivatives
    • If fx(x,y) and fy(x,y) are differentiable functions, then
    • chart?chf=bg,s,00000000&cht=tx&chl=f_%7Bxx%7D(x%2Cy)%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20x%7D%5Bf_x(x%2Cy)%5D%3D%5Cfrac%7B%5Cpartial%20f(x%2Cy)%7D%7B%5Cpartial%20x%5E2%7D&chs=496x78
    • chart?chf=bg,s,00000000&cht=tx&chl=f_%7Byy%7D(x%2Cy)%3D%5Cfrac%7B%5Cpartial%7D%7B%5Cpartial%20y%7D%5Bf_y(x%2Cy)%5D%3D%5Cfrac%7B%5Cpartial%20f(x%2Cy)%7D%7B%5Cpartial%20y%5E2%7D&chs=488x84
  14. Differential (Total Differential)
    • Let z=f(x,y) be a differential function then
    • dz=fx(x,y)dx+fy(x,y)dy is the total differential of z=f(x,y)
  15. An exact DE
    • A fist order linear or nonlinear d.e. of the form
    • M(x,y)dx+N(x,y)dy=0 or M(x,y)+N(x,y)dy/dx=0 is said to be an exact d.e. if there exists a function f(x,y) defined on some region R of the xy-plane such that chart?chf=bg,s,00000000&cht=tx&chl=M(x%2Cy)dx)dy%2BN(x%2Cy)%3Df_x(x%2Cy)dx%2Bf_y(x%2Cy)dy&chs=726x44
  16. Integrating Factor
    • The function μ(x) or μ(y) that when multiplied with a non exact d.e. M(x,y)dx+N(x,y)dy=0 converts it into an exact d.e.
    • The resulting d.e. may not be equivalent to the original but the solution of one is also the solution of the other.
  17. Homogeneous Function
    A function f(x,y) is said to be a homogeneous function of degree n≥0∈ℝ if f(tx,ty)=tn[f(x,y)] for a non-zero constant t or t≠0
  18. Homogeneous D.E.
    A d.e. of the form chart?chf=bg,s,00000000&cht=tx&chl=M(x%2Cy)dx%2BN(x%2Cy)dy%3D0&chs=384x38 is said to be a homogeneous d.e. if M(x,y) and N(x,y) are homogeneous functions of the same degree n≥0∈ℝ
  19. Bernoulli's DE
    • A differential equation of the form
    • chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%20%7Bdy%7D%7Bdx%7D%2BP(x)y%3Df(x)y%5En&chs=320x74 where y≠0 and n∈ℝ
  20. Separable in Variable
    A first order in the form chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%7Bdy%7D%7Bdx%7D%3Df(x%2Cy)&chs=176x74. If f(x,y) can be expressed as a product of g(x) and h(y), then chart?chf=bg,s,00000000&cht=tx&chl=%5Cfrac%7Bdy%7D%7Bdx%7D%3Df(x%2Cy)&chs=176x74 is separable in variable, or chart?chf=bg,s,00000000&cht=tx&chl=f(x%2Cy)%3Dg(x)h(y)&chs=260x38
Card Set
D.E. Definitions
Definitions for Differential Equations course