This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Review Chapters 1 to 5.7 Chapter 1  Introduction 1. Differential Equation : derivative and an equal sign. 2. Dependent Variable : varies as a function of another variable. 3. Independent Variable : does not vary with any other variables. 4. Ordinary Differential Equation : only 1 independent variable. 5. Partial Differential Equation : more than 1 independent variable; distinguishable by use of partial derivatives. 6. Order : highest derivative value in an equation. 7. Linear : dependent variable and its derivatives only have powers of 1. No constraints exist for independent variable(s). 8. Nonlinear : trig, exponentials, transcendental functions, multiplications, and divisions of depen dent variable and/or its derivatives. 9. Direction Field : Graph of slope using arrows. Shows the flow of solutions and the particular path of a solution for a given initial condition. 10. Implicit Solution : Answer of a differential equation has mixed variables. 11. Explicit Solution : Answer of a differential equation is of form y = f ( t ). Chapter 2  1st Order Ordinary Differential Equations 1. Linear Equations : dy dt + p ( t ) y = q ( t ). The integrating factor = e R p ( t ) dt which is multiplied by every term in the original equation, reduces the equation to d dt ( y ) = Q , such that solving for y yields y = 1 R qdt + C . 2. Separable Equations : dy dt = g ( t ) p ( y ). Separate variables and differentials of t on one side of the equal sign, those for y on the other, and integrate both sides to get an implicit equation for y . Solve for y if possible. 3. Exact Equations : d ( x,y ) = x dx + y dy = Mdx + Ndy = 0. Solutions are given implic itly by the level curve ( x,y ) = C , whose total derivative must be zero (derivative of constant C ). To test for exactness, the second partials of must be equal (since the order of taking two partials is commutative), so y x = x y or equivalently, M y = N x must hold. is then given by: = R Mdx + h ( y ) = C , where h ( y ) = R [ N y ( R Mdx )] dy , 1 or = R Ndy + g ( x ) = C , where g ( x ) = R [ M x ( R Ndy )] dx , by using the established relationships from the total derivative, that x = M and y = N , and integrating to solve for and the function of integration h ( y ) or g ( x ). Special Integrating Factors : Mdx + Ndy = 0 is exact. When an equation is not exact, a special integrating factor multiplied by every term in the nonexact equation may make it exact by satisfying the exactness test y ( M ) = x ( N ). We assume the integrating factor is only a function of x or y . In solving for the exactness test, is given by: ( x ) = e R Adx , if A = ( M y N x ) /N is only a function of x , else ( y ) = e R Bdx , if B = ( M y N x ) /M is only a function of y ....
View Full
Document
 Spring '08
 Fonken
 Calculus, Derivative

Click to edit the document details