4
Slope Fields, Solution Curves, and Eulers Method
You may recall from calculus that it is not always possible to evaluate an integral
symbolically. For example, the integral
Here the term elementary functions
refers to polynomials, exponentials,
logarith
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3
Applications of Dierential Equations
Dierential equations are absolutely fundamental to modern science and engineering.
Almost all of the known laws of physics and chemistry are actually dierential equations, and dierential equation models are used exte
Derivatives & Integrals Review
Chain Rule
Find the derivative of the given function.
1. f (x) = sin(x5 )
4. F (x) =
1
1 + x2
2. g (x) = (ln x)3/2
3. h(t) = cos2 t
5. k (y ) = ln(1 ey )
6. G(x) = e
x
Product Rule
Find the derivative of the given function.
2
Integrating Both Sides
So far, the only general method we have for solving dierential equations involves
equations of the form
y = f (x),
Of course, we also know how solve
equations of the form y = f (x) by
integrating twice.
where f (x) is any function
1
Introduction to Dierential Equations
A dierential equation is an equation that involves the derivative of some unknown
function. For example, consider the equation
f (x) = 4x3 .
(1)
This equation tells us information about the derivative f (x) of some f