Calculus Volume 2 · 免费预览
一阶线性方程
一阶线性方程 y 的导数加上 x 的函数 p 乘以 y 等于 x 的函数 q，可通过积分因子求解：这种乘子能把左端化为某个乘积的导数，积分后即得通解。
1. Standard form and the integrating factor
Standard form and the integrating factor
2. The homogeneous-plus-particular structure
The homogeneous-plus-particular structure
3. Initial value problems
Initial value problems
4. Applications
Applications
5. Standard form
To apply the integrating factor method, a linear equation must first be written in which form?
6. Integrating factor formula
For y prime plus p of x y equals q of x, the integrating factor is which function?
7. Collapse mechanism
Multiplying y prime plus p y by the integrating factor produces which left side?
8. Homogeneous role
In the general solution of a linear equation, the homogeneous term C e^(-integral p) is also called what?
9. Solvable case completeness
The chapter has solved separable and linear first-order equations. An equation like y prime = x + y is which type?
10. Integrating factor for 2y
For y prime + 2y = q, the integrating factor is e to the integral of 2 dx, which is e to which multiple of x? Answer like 2x.
11. Particular of constant forcing
For y prime + 2y = 3, the steady-state particular solution is 3/2. As a fraction, what is it? Answer like 3/2.
12. RC steady charge
An RC circuit charges toward Q = C V. As a two-letter symbol, what is the steady charge? Answer like Q with a subscript described.
13. Bernoulli preview
The equation y prime + p y = q y^n is called Bernoulli's equation, nonlinear for n not 0 or 1. Which substitution linearizes it?
14. Transient decay rate
The homogeneous transient decays at rate p. For p = 2, the transient halves in time ln 2 over 2. As a fraction, what is the half-life? Answer like ln 2/2.