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Differential equation

Algebra Fill In Notes 2.7

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02.07 Literal Equations Essential Questions How can you solve linear equations and inequalities in one variable, including equations with coefficients represented by letters? How can you rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations? In the equation: "distance equals rate times time? ____________________ is the most important part. d=(r)(t) In d=(r)(t) the more efficient way to evaluate r is to__________________ it from the rest of the equation first. Work ______________________ to isolate the variable. The key to equations is ________________________.

Ch9 SG

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372 CHAPTER 9 Mathematical Modeling with Differential Equations EXERCISE SET 9.1 1. y? = 2x2ex 3/3 = x2y and y(0) = 2 by inspection. 2. y? = x3 ? 2 sinx, y(0) = 3 by inspection. 3. (a) ?rst order; dy dx = c; (1 + x) dy dx = (1 + x)c = y (b) second order; y? = c1 cos t? c2 sin t, y?? + y = ?c1 sin t? c2 cos t+ (c1 sin t+ c2 cos t) = 0 4. (a) ?rst order; 2 dy dx + y = 2 ( ? c 2 e?x/2 + 1 ) + ce?x/2 + x? 3 = x? 1 (b) second order; y? = c1et ? c2e?t, y?? ? y = c1et + c2e?t ? ( c1et + c2e?t ) = 0 5. 1 y dy dx = x dy dx + y, dy dx (1? xy) = y2, dy dx = y2 1? xy 6. 2x+ y2 + 2xy dy dx = 0, by inspection. 7. (a) IF: ? = e3 ? dx = e3x, d dx [ ye3x ] = 0, ye3x = C, y = Ce?3x separation of variables: dy y = ?3dx, ln |y| = ?3x+ C1, y = ?e?3xeC1 = Ce?3x

Solving Systems by Substitution

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A good way to solve systems of equations is by substitution. In this method, you solve on equation for one variable, then you substitute that solution in the other equation, and solve. Example: 1. Problem: Solve the following system: x + y = 11 3x - y = 5 Solution: Solve the first equation for y (you could solve for x - it doesn't matter). y = 11 - x Now, substitute 11 - x for y in the second equation. This gives the equation one variable, which earlier algebra work has taught you how to do. 3x - (11 - x) = 5 3x - 11 + x = 5 4x = 16 x = 4

Linear Equations

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Linear Equations A linear equation in one variable is any equation of that may be written in the form Ax + B = C where A, B, and C are real number coefficients and x represents any real number that is a solution. Another way to define a linear equation in one variable is that the equation may always be written so that all the terms are either constants or some multiple of the variable raised to only the first power. Examples: 3(x - 3) = 2x + 5 is a linear equation since when we multiply out the left side we get the equation 3x - 9 = 2x + 5. 3/x + 4 = 5 is not a linear equation since the term 3/x is the same as 3x-1, a term with power other than 1. Method To Solve Linear Equations: To solve linear equations, remember to do the following:

Implicit Differentiation Simplified

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y + xy = 7x Implicit differentiation? (dy/dx) + (x(dy/dx)+ y(1)) = 7(1) (dy/dx)(x + 1) = 7 - y (dy/dx) = (7 - y) / (x + 1) Easy?
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