微积分教学资料——chapter9

For example, the function (2) y(t) = -16t2 + 40t +144 is a particular solution of Equation
d2y 32 2 dt
which satisfies the side conditions y(0) = 144, y’(0) = 40.
The initial-value problem f ( x) 2 x f (1) 2
Initial condition or side condition
y
Thus f(x) 2 xdx x 2 C
Using the fact that f(1)=2
y x2 1
Solution We differentiate the expression for y:
cet (1 cet ) (1 cet )(cet ) 2cet y t 2 (1 ce ) (1 cet ) 2
2
The right side of the differential equation becomes
y 1 dy (x 1)dx y Now,
y0
y 1 y dy (x 1)dx, 1 2 y ln y x x C 2
The function y=0 is also a solution of the given differ -ential eqution
Thus, the solution of the initial-value problem is
y 3 x3 8
Example 2 Find a solution of the differential equation xy y y , y1
which satisfies the side condition y(2) = 1. Solution We show first that the equation is separable:
are equations of the second-order.
A function f is called a solution of a differential equation if the equation is satisfied when y=f(x) and its derivatives are substituted into the equation.
is the general solutions which defines y implicitly as a function of x. To find a solution which satisfies the side condition, we set x = 2 and y = 1 in the oneparameter family: 1 + ln1 = 1/2×22 – 2 + C and C = 1. Thus, the particular solution is y ln y 1 x 2 x 1
*9.5 The Logistic Equation
9.6 Linear Equations
9.1 Modeling with Differential equations
Models of Population Growth Let t =time, P =the number of individuals in the population. Our assumption that the rate of growth of the population is proportional to the population size is written as the equation
dy f ( x)dx g ( y) (provided g ( y ) 0)
because f(x) and g(y) are continuous, they have antiderivatives. Thus, we get
1 g ( y) dy f ( x)dx
This is the general solution of the equation.
dP kP dt (1)
where k is the proportionality constant.
Example Find a function f such that its graph passes through the point（1，2），and that the slope of its tangent line at (x,f(x)) is 2x. Solution
x xy y e , y xy yy y sin x, 2u 2u 2 0 ( Laplace' s equation). 2 x y
The order of a differential equation is the order of the highest derivative (of the unknown func
For example, we know that the general solution of 3 the differential equation y x is given by
y 1 4 x C 4
Hale Waihona Puke where C is an arbitrary constant.
Example 1 Show that every member of the family of functions 1 cet y 1 cet is a solution of the differential equation y 1 ( y 2 1).
dy x 2 Example 1 (a) Solve the differential equation 2 . dx y
(b) Find the solution of this equation that satisfies the initial condition y(0) 2.
Solution (a) Note that this equation can be written as
2
Example 2 Find a one-parameter family of solution of the equation
dy xy dx
Solution Note that this equation can be written as dy xdx, y 0 y 1 2 so ln y x c1 2
y ec1 e
1 2 x 2
ce
1 2 x 2
where
c e 0
c1
The function y=0 is also a solution of the given differ -ential eqution Thus the general solutions is
y ce
1 2 1 1 cet 2 2cet ( y 1) [( ) 1] t 2 2 1 ce (1 cet ) 2
Therefore, for every value of c, the given function is a solution of the differential equation. c is often called parameter. The set of solutions of the second-order differential equation 2 is given by
-tion) that appears in the equation.
P(t ) kP(t ), y xy, xy 2 y x 2
are first-order equations;
x 2 y xy x 2 y 0, 2u 2u 2 2 x y
d y 32 2 dt (1)
y(t) = -16t2 + C1t + C2, (2) where C1 and C2 are independent arbitrary constants. This equation (1) has a two-parameter family of solutions.
2 1 1 x
We have C 1
A general solution o
So f(x) x 1
2
A particular solution
General Differential Equations
A differential equation is an equation that contains an unknown function and one or more of its derivatives.
y 2 dy x 2 dx
so
2 2 y dy x dx
1 3 1 3 y x C 3 3
where C is an arbitrary constant.
(b) If we put y(0) =2 in the general solution in part (a), we have C= 8/3.
1 2 x 2
where C is an arbitrary constant.
9.3 Separable Equations ; Homogeneous Equations The first-order equation is separable iff it can be put in the form y’ = f(x)g(y) where f(x) and g(y) are continuous. To solve such an equation, we begin by writing as
Chapter 9 Differential Equations
9.1 Modeling with Differential Equations
*9.2 Direction Fields and Euler’s Method
9.3 Separable Equations
*9.4 Exponential Growth and Decay
Thus, an nth-order equation has an nth-parameter family of solutions. Generally, the term general solution is often used in place of nth-parameter family of solutions. If specific values are assigned to each of the arbitrary constants in an nth-parameter family of solutions, then the resulting solution is called a particular solution. Specific values for constants are usually determined by imposing additional conditions, called side conditions or initial conditions, and the problem of finding a solution of the differential eqution that satisfies the initial condition is called an initial-value problem.