Chapter 2: First-Order Differential Equations Part 1 Department of - PowerPoint PPT Presentation

Overview Solution Curves without a Solution A Numerical Method Separable Equations Linear Equations Summary Chapter 2: First-Order Differential Equations – Part 1 Department of Electrical Engineering National Taiwan University ihwang@ntu.edu.tw September 17, 2013 DE Lecture 2 王奕翔 王奕翔

Overview 4 Separable Equations 6 Summary Solutions/Functions Defined by Integrals Discontinuous Coefficients Method 5 Linear Equations 3 A Numerical Method Solution Curves without a Solution 2 Solution Curves without a Solution 1 Overview Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 王奕翔

Overview 4 Separable Equations 6 Summary Solutions/Functions Defined by Integrals Discontinuous Coefficients Method 5 Linear Equations 3 A Numerical Method Solution Curves without a Solution 2 Solution Curves without a Solution 1 Overview Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 王奕翔

Overview Solution Curves without a Solution A Numerical Method Separable Equations Linear Equations Summary First-Order Differential Equation Throughout Chapter 2, we focus on solving the first-order ODE: Problem dy (1) DE Lecture 2 Find y = φ ( x ) satisfying dx = f ( x , y ) , subject to y ( x 0 ) = y 0 王奕翔

Overview Solution Curves without a Solution Fourier Transform (14) Fourier Series (11) Laplace Transform (7) 5 Transformation 4 Series Solution (6) Solutions by Substitutions (2-5): Solving Exact Equations (2-4) Solving Linear Equations (2-3) Separable Equations (2-2) Take antiderivative ( Calculus I, II ) 3 Analytic Method 2 Numerical Method (2-6, 9) 1 Graphical Method (2-1) Methods of Solving First-Order ODE Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 homogeneous equations, Bernoulli’s equation, y ′ = Ax + By + C . 王奕翔

Overview Organization of Lectures in Chapter 2 and 3 Solution Curves without a Solution We will not follow the order in the textbook. Instead, DE Lecture 2 Summary A Numerical Method Separable Equations Linear Equations ��� Separable DE (2-2) ��� �� DE Linear (2-1) (2-3) Models (3-1) ���� Exact DE Nonlinear (2-6) (2-4) Models (3-2) ���� (2-5) 王奕翔

Overview 4 Separable Equations 6 Summary Solutions/Functions Defined by Integrals Discontinuous Coefficients Method 5 Linear Equations 3 A Numerical Method Solution Curves without a Solution 2 Solution Curves without a Solution 1 Overview Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 王奕翔

Overview Example 1 (Zill&Wright p.36, Fig. 2.1.1.) Solution Curves without a Solution dy DE Lecture 2 Summary Separable Equations Linear Equations A Numerical Method dx = 0 . 2 xy y y solution slope = 1.2 curv e (2, 3) (2, 3) tangent x x 王奕翔

Overview dy arrow indicating the direction of the tangent line. Hence, at every point on the xy -plane, one can in principle sketch an Solution Curves without a Solution dx DE Lecture 2 Key Observation Direction Fields Summary Linear Equations Separable Equations A Numerical Method On the xy -plane, at a point ( x n , y n ) , the first-order derivative � � � � x = x n is the slope of the tangent line of the curve y ( x ) at ( x n , y n ) . From the initial point ( x 0 , y 0 ) , one can connect all the arrows one by one and then sketch the solution curve. ( 土法煉鋼！ ) 王奕翔

Overview dy Figure : Family of Solution Curves Figure : Direction Field Solution Curves without a Solution DE Lecture 2 Example 1 (Zill&Wright p.37, Fig. 2.1.3.) A Numerical Method Linear Equations Separable Equations Summary dx = 0 . 2 xy y y 4 4 c>0 2 2 c=0 x x c<0 _2 _2 _4 _4 _4 _2 2 4 _4 _2 2 4 王奕翔

Overview Example 2 (Zill&Wright p.37-38, Fig. 2.1.4.) Solution Curves without a Solution dy DE Lecture 2 Summary Linear Equations Separable Equations A Numerical Method dx = sin y , y (0) = − 1 . 5 y 4 ( x 0 , y 0 ) = (0 , − 1 . 5) 2 x _2 _4 _4 _2 2 4 王奕翔

Overview 4 Separable Equations 6 Summary Solutions/Functions Defined by Integrals Discontinuous Coefficients Method 5 Linear Equations 3 A Numerical Method Solution Curves without a Solution 2 Solution Curves without a Solution 1 Overview Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 王奕翔

Overview y Increment: . . . . . . Second Point: Solution Curves without a Solution dy dx DE Lecture 2 x Increment: A Numerical Method Separable Equations Initial Point: Linear Equations Summary Euler’s Method The graphical method of “connecting arrows” on the directional field can be mathematically thought of as follows: ( x 0 , y 0 ) x 1 = x 0 + h ( ) � � y 1 = y 0 + h = y 0 + hf ( x 0 , y 0 ) � � x = x 0 ( x 1 , y 1 ) 王奕翔

Overview Euler’s Method Solution Curves without a Solution Recursive Formula DE Lecture 2 Summary Linear Equations Separable Equations A Numerical Method Let h > 0 be the recursive step size, x n +1 = x n + h , y n +1 = y n + hf ( x n , y n ) , ∀ n ≥ 0 x n − 1 = x n − h , y n − 1 = y n − hf ( x n , y n ) , ∀ n ≤ 0 王奕翔

Overview Summary Solution Curves without a Solution Illustration DE Lecture 2 Linear Equations Separable Equations A Numerical Method y Solution Curve y ( x ) ( x 1 , y 1 ) ( x 0 , y 0 ) x x 0 x 1 王奕翔

Overview Summary Solution Curves without a Solution Illustration DE Lecture 2 Linear Equations Separable Equations A Numerical Method y Solution Curve y ( x ) ( x 2 , y 2 ) ( x 1 , y 1 ) ( x 0 , y 0 ) x x 0 x 1 x 2 王奕翔

Overview Summary Solution Curves without a Solution Illustration DE Lecture 2 Linear Equations Separable Equations A Numerical Method y Solution Curve Numerical y ( x ) Solution Curve ( x 2 , y 2 ) ( x 1 , y 1 ) ( x 0 , y 0 ) x x 0 x 1 x 2 王奕翔

Overview Solution Curves without a Solution A Numerical Method Separable Equations Linear Equations Summary Remarks The approximate numerical solution converges to the actual solution Euler’s method is just one simple numerical method for solving differential equations. Chapter 9 of the textbook introduces more advanced methods. DE Lecture 2 as h → 0 . 王奕翔

Overview 4 Separable Equations 6 Summary Solutions/Functions Defined by Integrals Discontinuous Coefficients Method 5 Linear Equations 3 A Numerical Method Solution Curves without a Solution 2 Solution Curves without a Solution 1 Overview Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 王奕翔

Overview Solution Curves without a Solution special structure of it. We start by inspecting the equation and see if we can identify some (1) dy Problem Recall the first-order ODE (1) we would like to solve Solving (1) Analytically Summary Linear Equations Separable Equations A Numerical Method DE Lecture 2 Find y = φ ( x ) satisfying dx = f ( x , y ) , subject to y ( x 0 ) = y 0 王奕翔

Overview dy Method: Direct Integration Solution Curves without a Solution DE Lecture 2 Summary Linear Equations Separable Equations A Numerical Method When f ( x , y ) depends only on x If f ( x , y ) = g ( x ) , then by what we learn in Calculus I & II, ∫ x dx = g ( x ) = ⇒ y ( x ) = g ( t ) dt + y 0 x 0 In the first-order ODE (1), if f ( x , y ) = g ( x ) only depends on x , it can be solved by directly integrating the function g ( x ) . 王奕翔

Overview Solve Plugging in the initial condition, we have A: From calculus we know that the Solution Curves without a Solution dy DE Lecture 2 Example Summary Linear Equations Separable Equations A Numerical Method When f ( x , y ) depends only on x dx = 1 x + e x , subject to y ( − 1) = 0 . ∫ 1 ∫ xdx = ln | x | , e x dx = e x y ( x ) = ln | x | + e x − 1 e , x < 0 . 王奕翔

Overview dy Then, we have That is, dy Solution Curves without a Solution = integrate both sides DE Lecture 2 Separable Equations dy Summary A Numerical Method Linear Equations When f ( x , y ) depends only on y If f ( x , y ) = h ( y ) , then ∫ y dx = h ( y ) = ⇒ h ( y ) = dx ⇒ h ( y ) = x − x 0 y 0 Assume that the antiderivative ( 不定積分、反導函數 ) of 1/ h ( y ) is H ( y ) . ∫ 1 h ( y ) dy = H ( y ) . ⇒ y ( x ) = H − 1 ( x − x 0 + H ( y 0 )) H ( y ) − H ( y 0 ) = x − x 0 = 王奕翔

Overview A: Use the same principle, we have = = = Solution Curves without a Solution dy dy DE Lecture 2 Linear Equations Example A Numerical Method Separable Equations Summary When f ( x , y ) depends only on y dx = ( y − 1) 2 . Solve dy dx = ( y − 1) 2 = ⇒ ( y − 1) 2 = dx , y ̸ = 1 1 ⇒ 1 − y = x + c , for some constant c 1 ⇒ y = 1 − x + c , for some constant c , or y = 1 Note: How about the constant function y = 1 ? ⇒ y = 1 is called a singular solution . 王奕翔

Overview a u . . . . . . cot u Solution Curves without a Solution cos u sin u a u tan u DE Lecture 2 A Numerical Method Separable Equations Linear Equations u n Summary Antiderivative Table of Integrals Function u n +1 n + 1 + C , n ̸ = − 1 u − 1 ln | u | + C ln a + C − cos u + C sin u + C − ln | cos u | + C ln | sin u | + C 1 1 a tan − 1 u a + C a 2 + u 2 1 sin − 1 u a + C √ a 2 − u 2 王奕翔