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| 1. | a. | x3dx
= -(y + 1)2dy 1/4x4 = -1/3(y + 1)3 + c 3x4 = -4(y + 1)3 + c |
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| b. | x2(y
+ 1)dx + y2(x - 1)dy = 0
x²/(x - 1) dx = -y²/(y + 1)dy (x + 1 + 1/(x - 1))dx = -(y - 1 + 1/(y + 1))dy (staartdeling maken) 1/2x2 + x + ln(x - 1) = -1/2y2 + y - ln(y + 1) + c x2 + 2x + 2ln(x - 1) + 2ln(y + 1) + y2 - 2y = c x2 + 2x + 2ln((x - 1)(y + 1)) + y2 - 2y = c misschien nog mooier: (x + 1)2 + (y - 1)2 + 2ln((x - 1)(y + 1)) = c (nieuwe c) |
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| c. | 4xdy -
ydx = x2dy dy(4x - x2) = ydx 1/y • dy = 1/(4x - x²) • dx 1/y • dy = (0,25/x + -0,25/(4 - x)) dx (breuksplitsen) lny = 1/4lnx - 1/4ln(4 - x) + c 4lny = ln(x/(4 - x)) + c ln(y4) = ln(x /(4 - x)) + c y4 = c • x /(4 - x) (nieuwe c) (4 - x)y4 = cx |
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| d. | x(y -
3)dy = 4ydx (y - 3)/y • dy = 4/x • dx (1 - 3/y)dy = 4/x • dx y - 3lny = 4lnx + c ey • y-3 = x4 • c (nieuwe c) ey = cx4y3 |
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| e. | cosy • dx
+ (1 + e-x)siny • dy = 0 (1/(1 + e-x)) • dx = -(siny/cosy) dy (ex/(ex + 1)) • dx = -(siny/cosy) dy ln(ex + 1) = ln(cosy) + c ex + 1 = c • cosy (of misschien x = ln(c • cosy - 1) als je dat mooier vindt) |
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| f. | (x2
+ 1) • dy = tany • dx 1/tany • dy = 1/(x² + 1) • dx primitiveren: ln(siny) = arctanx + c |
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| g. | dr +
r • tanφ • dφ
= 0 1/r • dr = -tanφ dφ lnr = ln(cosφ) + c r = c • cosφ |
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| h. | xyy' = (y +
1)(1 - x) y' • y/(y + 1) = (1 - x)/x dy • (1 - 1/(y + 1)) = dx • (1/x - 1) y - ln(y + 1) = lnx - x + c mooier: x + y = c + ln(x(y + 1)) |
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© h.hofstede (h.hofstede@hogeland.nl) |
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