First-Order Differential Equations S.-Y. Leu Sept. 28, 2005

CHAPTER 2 First-Order Differential Equations 2.1 Solution Curves Without the Solution 2.2 Separable Variables 2.3 Linear Equations 2.4 Exact Equations 2.5 Solutions by Substitutions 2.6 A Numerical Solution 2.7 Linear Models 2.8 Nonlinear Models 2.9 Systems: Linear and Nonlinear Models

2.1 Solution Curves Without the Solution x y Slope= Short tangent segments suggest the shape of the curve 輪廓

2.1 Solution Curves Without the Solution The general first-order differential equation has the form F(x, y, y’)=0 or in the explicit form y’=f(x,y) Note that, a graph of a solution of a first-order differential equation is called a solution curve or an integral curve of the equation. On the other hand, the slope of the integral curve through a given point (x0,y0) is y’(x0).

2.1 Solution Curves Without the Solution A drawing of the plane, with short line segments of slope drawn at selected points , is called a direction field or a slope field of the differential equation . The name derives from the fact that at each point the line segment gives the direction of the integral curve through that point. The line segments are called lineal elements.

2.1 Solution Curves Without the Solution Plotting Direction Fields 1st Step y’=f(x,y)=C=constant curves of equal inclination 2nd Step Along each curve f(x,y)=C, draw lineal elements  direction field 3rd Step Sketch approximate solution curves having the directions of the lineal elements as their tangent directions.

2.1 Solution Curves Without the Solution If the derivative dy/dx is positive (negative) for all x in an interval I, then the differentiable function y(x) is increasing (decreasing) on I.

2.1 Solution Curves Without the Solution DEFINITION: autonomous DE A differential equation in which the independent variable does not explicitly appear is known as an autonomous differential equation. For example, a first order autonomous DE has the form DEFINITION: critical point A critical point of an autonomous DE is a real number c such that f(c) = 0. Another name for critical point is stationary point or equilibrium point. If c is a critical point of an autonomous DE, then y(x) = c is a constant solution of the DE.

2.1 Solution Curves Without the Solution DEFINITION: phase portrait A one dimensional phase portrait of an autonomous DE is a diagram which indicates the values of the dependent variable for which y is increasing, decreasing or constant. Sometimes the vertical line of the phase portrait is called a phase line.

DEFINITION: Separable DE 2.2 Separable Variables DEFINITION: Separable DE A first-order differential equation of the form is said to be separable or to have separable variables. (Zill, Definition 2.1, page 44).

2.2 Separable Variables Method of Solution  If represents a solution

2.2 Separable Variables domain , , The Natural Logarithm The natural exponential function domain

DEFINITION: Linear Equation 2.3 Linear Equations DEFINITION: Linear Equation A first-order differential equation of the form is said to be a linear equation. (Zill, Definition 2.2, page 51). When homogeneous Otherwise it is non-homogeneous.

2.3 Linear Equations Standard Form 

2.2 Separable Equations A differential equation is called separable if it can be written as Such that we can separate the variables and write We attempt to integrate this equation

2.2 Separable Equations Example 1. is separable. Write as Integrate this equation to obtain or in the explicit form What about y=0 ? Singular solution !

2.2 Separable Equations Example 2. is separable, too. We write Integrate the separated equation to obtain The general solution is Again, check if y=-1 is a solution or not ? it is a solution, but not a singular one, since it is a special case of the general solution 

2.3 Linear Differential Equations A first-order differential equation is linear if it has the form Multiply the differential equation by to get Now integrate to obtain The function is called an integrating factor for the differential equation. 

2.3 Linear Differential Equations Linear: A differential equation is called linear if there are no multiplications among dependent variables and their derivatives. In other words, all coefficients are functions of independent variables. Non-linear: Differential equations that do not satisfy the definition of linear are non-linear. Quasi-linear: For a non-linear differential equation, if there are no multiplications among all dependent variables and their derivatives in the highest derivative term, the differential equation is considered to be quasi-linear.

2.3 Linear Differential Equations Example is a linear DE. P(x)=1 and q(x)=sin(x), both continuous for all x. An integrating factor is Multiply the DE by to get Or Integrate to get The general solution is

2.3 Linear Differential Equations Example Solve the initial value problem It can be written in linear form An integrating factor is for Multiply the DE by to get Or Integrate to get ,then for For the initial condition, we need C=17/4 the solution of the initial value problem is