# 12.5 Differentiable Equations

## 12.5.2 Initial Value Problems

When you antidifferentiate, you need to do a substitution to find the function

## 12.5.6 Newton's Law of Cooling

Stuff tries to become closer to the temperature around it.

## 12.5.3 Integration by Substitution

Give stuff a nickname, it makes it easier to work with them.

## 12.5.5 Exponential Growth and Decay

When there is more stuff, a larger amount grows than when there is less stuff.  It's a rate problem

Same with decay, but the reverse.

## 12.5.4 Separable Differential Equations

When a derivative is expressed with both variables, it is very likely to have been differentiated using implicit differentiation.  It becomes important to separate the variables and differentiate each side separately.  Then, you may, or may not be able to write the resulting equation using function notation, depending on the equation itself.

Resources

• Warmup: indefinite integrals
• Notes: Separable differential equations

Assignment:

• p322 #39-44

Things you should be able to do after today:

• integrate derivatives that are expressed in terms of both x and y
• use separable differentiable equations to integrate derivatives  in terms of both x and y where substitution may be needed