微分方程式的分类.ppt
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1、Chapter 1 First-Order Differential EquationsA differential equation defines a relationship between an unknown function and one or more of its derivativesApplicable to:ChemistryPhysicsEngineeringMedicineBiologyY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung微分
2、方程式的分類Ordinary Differential Equation:(ODE)y 只與一個變數 x 有關Partial Differential Equation:(PDE)u 與兩個變數 x,y 有關Total Differential Equation:(TDE)Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungFirst order differential equat
3、ion with y as the dependent variable and x as the independent variable would be:Second order differential equation would have the form:常微分方程式常微分方程式An ordinary differential equation is one with a single independent variable.The order(階階)of an equation:The order of the highest derivative appearing in
4、the equationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungLinear if the nth-order differential equation can be written:an(t)y(n)+an-1(t)y(n-1)+.+a1y+a0(t)y=h(t)Nonlinear not linearExample of an Ordinary Differenti
5、al EquationIf f(x)=0,The ODE is Homogeneous If f(x)0,The ODE is Non-homogeneous最高階導數的次數最高階導數的次數(degree)稱為此微分方程式之次數稱為此微分方程式之次數Example:Degree=1Degree=2Chapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Or
6、der Differential Equations初值問題與邊界問題初值問題與邊界問題Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungGeneral Solution all solutions to the differential equation can be represented in this form for all constantsParticular Solution contains no arbitrary constants()()()(
7、sincossolutionparticulardefinediscIfsolutiongeneralcxyxy+=Chapter 1 First-Order Differential Equations通解與特殊解通解與特殊解Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung奇異解奇異解(Singular Solutions)A differential equation may sometimes have an additional solution that
8、cannot be obtained from the general solution.solutionaalsoisxyparabolabutccxysolutiongeneralwithyyxy40222=-=+-Chapter 1 First-Order Differential Equations顯式解顯式解(explicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung可分離微分方程式可分離微分方程式(Separable Diffe
9、rential Equations)Special form分離變數分離變數(Separable Variables)Example:Boyles gas lawChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungExample 1:Solve the differential equationChapter 1 First-Order Differential Equations原
10、式原式:兩邊作積分兩邊作積分Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung試求試求:之通解之通解 (83清大化工清大化工)先利用座標平移來消去常數先利用座標平移來消去常數 2 與與 6 令令Chapter 1 First-Order Differential Equations隱式解隱式解(implicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National Univ
11、ersity of KaohsiungChapter 1 First-Order Differential EquationsReduction to Separable Form1.Differential equations of the form-有時稱為齊次有時稱為齊次(homogeneous)方程式方程式其中其中g為為y/x的任意函數的任意函數,例如例如:,欲求其解可令欲求其解可令 andY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1
12、First-Order Differential EquationsReduction to Separable Form2.Transformations Example:令令代入並簡化代入並簡化乘以乘以2並分離變數並分離變數積分可得積分可得隱式解隱式解(implicit solution)Y.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Pro
13、fessor,Department of Applied Physics,National University of KaohsiungFirst-Order Ordinary Differential EquationsFor(x,y)=CIfWe call Exact(正合正合)Differential EquationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of Kaohsiung試求試
14、求:之解之解 (84 (84中央光電中央光電)Exact Differential EquationChapter 1 First-Order Differential EquationsY.M.Hu,Assistant Professor,Department of Applied Physics,National University of KaohsiungInexact Differential EquationExact Differential EquationIfBut ifInexact Differential EquationAn integrating factorCha
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