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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 μ = e R p ( t ) dt which is multiplied by every term in the original equation, reduces the equation to d dt ( μy ) = μQ , such that solving for y yields y = 1 μ R μqdt + 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 : dψ ( x,y ) = ∂ψ ∂x dx + ∂ψ ∂y dy = Mdx + Ndy = 0. Solutions are given implic itly by the level curve ψ ( x,y ) = C , whose total derivative must be zero (derivative of constant C ). To test for exactness, the second partials of ψ must be equal (since the order of taking two partials is commutative), so ∂ ∂y ∂ψ ∂x = ∂ ∂x ∂ψ ∂y or equivalently, ∂M ∂y = ∂N ∂x must hold. ψ is then given by: ψ = R Mdx + h ( y ) = C , where h ( y ) = R [ N ∂ ∂y ( R Mdx )] dy , 1 or ψ = R Ndy + g ( x ) = C , where g ( x ) = R [ M ∂ ∂x ( R Ndy )] dx , by using the established relationships from the total derivative, that ∂ψ ∂x = M and ∂ψ ∂y = N , and integrating to solve for ψ and the function of integration h ( y ) or g ( x ). Special Integrating Factors : μMdx + μNdy = 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 ( μM ) = ∂ ∂x ( μN ). We assume the integrating factor is only a function of x or y . In solving for the exactness test, μ is given by: μ ( x ) = e R Adx , if A = ( ∂M ∂y ∂N ∂x ) /N is only a function of x , else μ ( y ) = e R Bdx , if B = ( ∂M ∂y ∂N ∂x ) /M is only a function of y ....
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 Spring '08
 Fonken
 Calculus, Derivative, Euler

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