341h1 - M341 H1(S Zhang 1 Differential equations and...

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M341 H1 (S. Zhang) . 1. Differential equations and mathematical models. § 1.1: 13, 15, 24, 26, 28, 33,35 ans: 2. Find a solution of type y = e rx for the (homogeneous, constant coefficient) differential equation: y 00 + y 0 - 2 y = 0 ans: § 1.1 15. r 2 e rx + re rx - 2 e rx = 0 r 2 + r - 2 = 0 , r = 1 , - 2 We get two solutions y = e x and y = e - 2 x . 3. ans: 4. Verify the given function satisfies the differential equation, then find the constant. y = x 3 ( C + ln x ) . xy 0 - 3 y = x 3 , y (1) = 17 ans: § 1.1:24. x (3 x 2 ( C + ln x ) + x 3 1 x ) - 3 x 3 ( C + ln x ) x 3 = x 3 17 = 1 3 ( C + 0) y = x 3 (17 + ln x ) 5. Verify the given function satisfies the differential equation, then find the constant. y = ( x + C ) cos x. y 0 + y tan x = cos x, y ( π ) = 0 ans: § 1.1:26. (cos x - ( x + C ) sin x ) + ( x + C ) sin x = cos x cos x = cos x 0 = ( π + C ) · ( - 1) , C = - π y = ( x - π ) cos x 6. Find a differential equation y 0 = f ( x, y ) so that a solu- tion y = g ( x ) has the described geometric property for its graph. (a) The line tangent to the graph of g at ( x, y ) intersects the x -axis at the point ( x/ 2 , 0). ans: (a) The tangent line goes through two points ( x, y ) and ( x/ 2 , 0). Therefore, its slope is (the ratio of y incre- ment over x increment) m = 0 - y ( x/ 2) - x y 0 = 2 y x 7. Model the problem by a differential equation. (a) The time rate of change of the velocity v of a boat is proportional to the square of v . ans: (a) § 1.1:33. dv dt = kv 2 8. Model the problem by a differential equation. (a) In a city with a fixed population of P persons, the time rate of change of the number N of those persons who have heard a certain rumor is proportional to the number of those who have not yet heard the rumor. ans: (a) § 1.1:35. dN dt = k ( P - N ) 9. Integrals as general and particular solutions. § 1.2: 5, 14, 19, 25, 27, 30, 36.
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