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Separation of variables

AP Calc AB 2004 Scoring Guideliens

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AP? Calculus AB 2004 Scoring Guidelines The College Board is a not-for-profit membership association whose mission is to connect students to college success and opportunity. Founded in 1900, the association is composed of more than 4,500 schools, colleges, universities, and other educational organizations. Each year, the College Board serves over three million students and their parents, 23,000 high schools, and 3,500 colleges through major programs and services in college admissions, guidance, assessment, financial aid, enrollment, and teaching and learning. Among its best-known programs are the SAT?, the PSAT/NMSQT?, and the Advanced Placement Program? (AP?). The College Board is committed to the principles of

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
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