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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. Non-linear : 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 non-exact 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 ....
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