Calculus Volume 2 · 免费预览
可分离方程
可分离方程可分解为关于 x 的函数与关于 y 的函数的乘积；分离变量后，将每个变量与其微分放在一起直接积分，从而通过两次积分求解方程。
1. The separation method
The separation method
2. Worked forms
Worked forms
3. Modeling with separation
Modeling with separation
4. The scope of separation
The scope of separation
5. Separable criterion
A differential equation is separable when the right side can be written in which form?
6. Separation step
To solve dy/dx = x y, which separated form results?
7. Implicit solutions
Separation of dy/dx = x/y gives y squared over 2 = x squared over 2 + C. If y cannot be conveniently isolated, what is the status of this relation?
8. Mixing steady state
A well-stirred tank's salt amount y(t) follows dy/dt = 20 - y/50. As t approaches infinity, y(t) approaches what value?
9. Lost solutions from division
Separating dy/dx = y(1 - y) by dividing by y(1 - y) loses which solutions?
10. Separated integral left side
Separating dy/dx = ky gives dy/y = k dx. Integrating the left side gives ln|y|. As a single expression, what is it? Answer like ln|y|.
11. Cooling as separation
Newton's cooling dT/dt = k(T - T_env) separates to dT/(T - T_env) = k dt. What solution form results?
12. Constant solution
For dy/dx = y(1 - y), the values y = 0 and y = 1 make the right side zero, giving constant solutions called what? Answer with one word.
13. Mixing separation
The mixing equation dy/dt = 20 - y/50 rewrites as dy/(1000 - y) = dt/50 after factoring. As a fraction, what is the coefficient of y in the denominator's form 1000 - y? Answer with a number.
14. Separation validity
Dividing by h(y) during separation is invalid where h(y) = 0, possibly losing solutions. The constant solutions so lost are recovered by what? Answer with two words.