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Separation of Variables Worksheet

Separation of Variables Worksheet
  • Page 1
 1.  
Solve the equation dydx = 3(y2-xy)xy-x2.
a.
(yx)12 = C1 x
b.
(xy)12 = C1 x
c.
(xy)12 = C1 x
d.
(yx)12 = C1
e.
(yx) 2 = C1 x


Solution:

dydx = 3(y2-xy)xy-x2
[Write the equation.]

dydx = 3y(y-x)x(y-x) = 3yx
[Factor both numerator and denominator.]

Let v = yx , y = vx

dydx = v + xdvdx

v + x dvdx = 3v

x dvdx = 2v

12v dv = 1x dx

1 / 21v dv = 1x dx
[Integrate on both sides.]

1 / 2ln |v| = ln |x| + C

ln |v12| = ln |x| + C

ln |v12| = ln |C1 x|
[C = ln |C1|.]

(yx)12 = C1 x
[Substitute v = yx.]


Correct answer : (1)
 2.  
Solve the equation dydx = x+yx-y.
a.
ln | x2 + y2 | = C
b.
2 tan-1 (yx) = C
c.
tan-1 (yx) = ln |C1 (x2 + y2) |
d.
2 tan-1(yx) = ln |C1 (x2 + y2) |
e.
tan-1 (y) = C1 ln | x2 + y2 |


Solution:

dydx = x+yx-y
[Write the differential equation.]

dydx = 1+yx1-yx
[Divide numerator and denominator by x.]

Put v = yx, y = vx

dydx = v + xdvdx

v + x dvdx = 1+v1-v
[Put v = yx.]

x dvdx = 1+v-v+v21-v

1-v1+v2 dv = 1x dx

1-v1+v2 dv = 1x dx
[Integrate on both sides.]

11+v2 dv - v1+v2 dv = 1x dx

tan-1 (v) - 1 / 2 ln | 1 + v2 | = ln | x | + C

2 tan-1 (v) = ln |x2| + ln | 1 + v2 | + C
[Multiply both sides by 2.]

2 tan-1 (v) = ln |C1 x2(1 + v2) |
[C = ln C1.]

2 tan-1 (yx) = ln |C1x2(1 + y2x2) |
[Replace v with yx.]

2 tan-1 (yx) = ln |C1 (x2 + y2) |


Correct answer : (4)
 3.  
Solve the equation xy′ = x + y.
a.
y = x ln | C1x |
b.
y = 1x + C
c.
x = y ln | x | + C
d.
y = ln | C1x |
e.
y = x + C


Solution:

y ′ = x+yx
[Write the differential equation.]

dydx = 1+yx1 = 1 + yx
[Divide numerator and denominator by x.]

Put v = yx, y = vx

dydx = v + xdvdx

v + x dvdx = 1 + v

x dvdx = 1

dv = 1x dx
[Integrate on both sides.]

v = ln | x | + C

v = ln | C1x |
[C = ln C1.]

yx = ln | C1x |
[Replace v with yx.]

y = x ln |C1x |


Correct answer : (1)
 4.  
Solve the equation x dy - y dx = x2+y2dx.
a.
(y + x2+y2) = C1 x2
b.
(x + x2+y2) = C1 x2
c.
(y + x2+y2) = C1 x
d.
xy = c1
e.
(xy + x2+y2) = C1 x


Solution:

x dy - y dx = x2+y2dx
[Write the differential equation.]

dydx = y+x2+y2x
[Find dydx.]

dydx = yx+1+y2x21
[Divide both numerator and denominator by x.]

Put v = yx, vx = y

v + x dvdx = dydx = v + 1+v2

x dvdx = 1+v2

dv1+v2 = 1x dx

dv1+v2 = 1x dx
[Integrate on both sides.]

ln | v + 1+v2 | = ln | x | + C

ln | v + 1+v2 | = ln | C1x |
[C = ln C1.]

ln | yx + 1+y2x2 | = ln | C1x |
[Replace v with yx.]

(y + x2+y2) = C1x2


Correct answer : (1)
 5.  
Solve dydx = xyx2+2y2.
a.
y2 = 4 ln | y | + C
b.
x2 = 4 ln | y | + C
c.
y2 = y2 ln | y | + C
d.
x2 = 4y2 + C
e.
x24y2 = ln | y | + C


Solution:

dydx = xyx2+2y2

dydx = yx1+2y2x2
[Divide both numerator and denominator by x2.]

Put v = yx, vx = y

v + x dvdx = dydx = v1+2v2

x dvdx = v1+2v2 - v

x dvdx = v-v-2v31+2v2

x dvdx = - 2v31+2v2

1+2v2-2v3 dv = 1x dx
[Integrate on both sides.]

- 1 / 2v-3 dv - 1v dv = 1x dx

14v2 - ln | v | = ln | x | + C

14v2 = ln | vx | + C

x24y2 = ln | y | + C
[Replace v with yx.]


Correct answer : (5)
 6.  
Solve dydx = x2+xy+y2x2.
a.
tan-1 (xy) = ln | x | + C
b.
tan-1 (yx) = x + C
c.
tan-1 (y) = ln | x | + C
d.
tan-1 (yx) = ln | x | + C
e.
tan-1 (1x) = ln | x | + C


Solution:

dydx = x2+xy+y2x2

dydx = 1+yx+y2x21
[Divide both numerator and denominator by x2.]

Put v = yx, vx = y

v + xdvdx = dydx

v + x dvdx = 1 + v + v2

x dvdx = 1 + v2

11+v2 dv = 1x dx
[Integrate on both sides.]

tan-1 (v) = ln | x | + C

tan-1 (yx) = ln | x | + C
[Replace v with yx.]


Correct answer : (4)
 7.  
Solve dydx = yx+y2x2.
a.
x + ln | x | = C
b.
yx + ln | x | = C
c.
xy + ln | x | = C
d.
xy + x = C
e.
1y + ln | x | = C


Solution:

dydx = yx+y2x2
[Write the differential equation.]

Put v = yx, vx = y

v + xdvdx = dydx

v + x dvdx = v + v2

x dvdx = v2

1v2 dv = 1x dx

1v2 dv = 1x dx
[Integrate on both sides.]

- 1v + C = ln | x |

xy + ln | x | = C
[Replace v with yx.]


Correct answer : (3)
 8.  
Solve y ′ = yx - cosecyx.
a.
ln | x | - cos y = C
b.
ln | x | + sin (yx) = C
c.
cos (yx) - ln | x | = C
d.
ln | x | + cos (xy) = C
e.
cos (yx) + x = C


Solution:

y ′ = yx - cosec yx
[Write the equation.]

dydx = yx - cosec yx

Put v = yx, vx = y

v + xdvdx = dydx

v + x dvdx = v - cosec v

x dvdx = - cosec v

- sin v dv = 1x dx
[Integrate on both sides.]

cos v = ln | x | + C

cos (yx) - ln | x | = C
[Replace v with yx.]


Correct answer : (3)
 9.  
Solve the homogeneous differential equation x dydx = y + x cos2 (yx).
a.
y = x ln | C1x |
b.
y = x tan-1 (ln | C1x |)
c.
x = y tan-1 (ln |C1x |)
d.
y = 1x tan-1 (ln | C1x |)
e.
y = tan-1 (ln | C1x |)


Solution:

x dydx = y + x cos2 (yx)
[Write differential equation.]

dydx = yx + cos2 (yx)

Put v = yx, vx = y

v + xdvdx = dydx

v + x dvdx = v + cos2 (v)

1cos2v dv = 1x dx

sec2v dv = 1x dx
[Integrate on both sides.]

tan v = ln | x | + C

tan (yx) = ln | C1x |
[C = ln C1 and Replace v with yx.]

y = x tan-1 (ln | C1 x |)


Correct answer : (2)
 10.  
Solve the homogeneous differential equation dydx = yx-e-yx.
a.
exy = 1x + C
b.
e1x = ln | C1x |
c.
ey = ln | C1x |
d.
eyx = ln | C1x |
e.
exy = ln | C1x |


Solution:

dydx = yx-e-yx
[Write the differential equation.]

Put v = yx, vx = y

v + xdvdx = dydx

v + x dvdx = v - e-v

x dvdx = - e-v

ev dv = - 1x dx

ev dv = - 1x dx
[Integrate both sides.]

ev = - ln | x | + C

ev = ln | 1x | + C

ev = ln | C1x |
[C = ln C1.]

eyx = ln | C1x |
[Replace v with yx.]


Correct answer : (4)

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