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© h.hofstede (h.hofstede@hogeland.nl)

cosx • ^dy/_dx = cos²x - sinx • y
^dy/_dx = cosx - ^sinx/_cosx • y
^dy/_dx + ^sinx/_cosx • y = cosx
f = ^sinx/_cosx = tanx dus   ∫ f = -ln(cosx)
h = e ^{∫ f} = e^-ln(cosx) = (cosx)^-1
hg = (cosx)^-1 • cosx = 1   dus ∫ hg = x
y = ^{(c + x)}/_(cosx)^-1 = c • cosx + xcosx

xy' = 2y + 3x⁴ + 2
y ' = ^2y/_x + 3x³ + ²/_x
y' - y • ²/_x = 3x³ + ²/_x
f = -²/_x dus ∫ f = -2lnx
h = e^{∫ f} = e^-2lnx = x^-2
hg = x^-2•(3x³ + ²/_x) = 3x + 2x^-3 dus ∫ hg = 1¹/₂x² - x^-2
y = (c + 1,5x² - x^-2)/x^-2 = cx² + 1¹/₂x⁴ - 1

^dy/_dx + 2xy = 2x
f = 2x dus ∫ f = x²
h = e^{∫ f} = e^x^²
hg = 2xe^x^² dus ∫ hg = e^x^²
y = (c + e^x^²) / e^x^² = ce^-x² + 1

2x• ^dy/_dx + y = 2x√x
^d^y/_d_x + ¹/₂_x • y = √x
f = ¹/_2x = ¹/₂ • ¹/_x dus ∫ f = ¹/₂lnx
h = e^{∫ f} = e^0,5lnx = √x
hg = √x • √x = x dus ∫ hg = ¹/₂x²
y = ^{(c + 0,5x²)}/_√_x = ^c/_√_x+ ¹/₂x√x

y' - 2y = e^3x
f = -2 dus ∫ f = -2x
h = e^{∫ f} = e^-2x
hg = e^-2x • e^3x = e^x dus ∫ hg = e^x
y = (c + e^x) / e^-2x = c • e^2x + e^3x

x² y' + xy = x² + 1
y ' + ^y/_x = 1 + ¹/_x²
f = ¹/_x dus ∫ f = lnx
h = e^∫^f = e^lnx = x
hg = x + ¹/_x dus ∫ hg = ¹/₂x² + lnx
y = (c + 0,5x² + lnx)/ x =^c/_x + ¹/₂x + ^lnx/_x

y' + ^y/_{(x - 1)} = ¹/_{(x - 1)(x + 3)}
f = ¹/_{(x- 1)} dus ∫ f = ln(x - 1)
h = e^{∫ f} = e^ln(x^{-
1)} = x - 1
hg = ¹/_{(x + 3)} dus ∫ hg = ln(x + 3)
y = ^{(c + ln(x + 3))}/_{(x -
1)}

(1 + x²) • y' + 2xy = 1 + x²
y' + ^2x/_{(1 + x²)} • y = 1
f = ^2x/_{(1 + x²)} dus ∫ f = ln(1 + x²)
h = e^{∫ f} = e^{ln(1 +
x²)} = 1 + x²
hg = 1 + x² dus ∫ hg = x + ¹/₃x³
y = (c + x + ¹/₃x³ )/(1 + x²)

y' x - 3y = 7x
y ' - ³/_x • y = 7
f = -³/_x dus ∫ f = -3lnx
h = e^{∫ f} = e^-3lnx = x^-3
hg = x^-3 • 7 dus ∫ hg = -⁷/₂x^-2
y = (c - ⁷/₂x^-2) / x^-3 = cx³ - ⁷/₂x

y' + 3y = e^-3x
f = 3 dus ∫ f = 3x
h = e^{∫ f} = e^3x
hg = e^3x• e^-3x = 1 dus ∫ hg = x
y = (c + x)/e^3x = (c + x) • e^-3x

xy' + 2y = 8x²
y ' + ²/_x • y = 8x
f = ²/_x dus ∫ f = 2lnx
h = e^{∫ f} = e^2lnx = x²
hg = x² • 8x = 8x³ dus ∫ hg = 2x⁴
y = (c + 2x⁴) / x² = cx^-2 + 2x²

xy' - 4y = x⁶e^x
y ' - ⁴/_x • y = x⁵e^x
f = -⁴/_x dus ∫ f = -4lnx
h = e^{∫ f} = e^-4lnx = x^-4
hg = x^-4 • x⁵e^x = xe^x dus ∫ hg = xe^x - e^x
y = (c + xe^x - e^x) / x^-4 = cx⁴ + x⁵e^x - x⁴e^x

cosx • y' + ysinx = 2cos³xsinx - 1
^dy/_dx + y sinx/cosx = 2cos²xsinx - ¹/_cos_x
f = ^sinx/_cosx = tanx dus ∫ f = -ln(cosx)
h = e^∫f = e^-^ln(cosx) = (cosx)^-1 hg = 2cosxsinx - ¹/_cos²x = sin2x - ¹/_cos²_x
∫ hg =-¹/₂cos2x - tanx
y = (c - ¹/₂cos2x - tanx) • cosx
y = c • cosx - ¹/₂cos2xcosx - tanxcosx