By Azizan & Faye, January 2014

Chapter 2

2.1.4 Separable

Solve the given differential equations by separation of variables.

1. 01 3 dx

dye x

2. 02sec dyxdx

3. y

x

dy

dx

4. yxe

dx

dy 5

5.

2

1

6

x

y

dx

dy

Solve the given differential equation subject to the indicated initial condition.

6. 23,1

)1(

xt

x

dt

dxt

7. 30,)1( 22 yxydx

dy

8. 10),1)(1( 2 yxydx

dy

9. 0,sin)12( yxydx

dy

10. dy

dx= 27x2e-3y, y 1( ) = 0

2.2.6 Linear

Find the general solution of the following:

1. 84 ydx

dy

2. 1sin32 xxxydx

dyx

By Azizan & Faye, January 2014

3. 32 ttwdt

dwt

4. 1cossin yd

dy

5. textdt

dxt 71

Solve the initial value problems, subject to the indicated initial conditions.

6. 12

,cos

Pdt

t

PttdP

7. 11,ln1 RRd

dR

8. 02,132 xtxdt

dxt

9. 01,2 22 yxyedx

dye xx

10. e

yyexdx

dyx x 1

1,2

2.3.5 Exact

For each of the following, determine whether the given DE is exact or non-exact. Solve each

exact equation.

1. dvuduvu 12

2. 03 duuedtuet tt

3. dyxyxdxyxx 3cossincos

4. 0cossinsin dyyxydxyx

5. 02

12

2

dv

v

uudu

vuv

6. 022 22 dyxyxdxxyy

7. 0dx

dyxy

8. dyyxdxyx )32()24(

By Azizan & Faye, January 2014

9. 0)()3( 222 dd

10. 0)2(2 dyyeydy y

2.4.3 Non- Exact

Prove that each of the given differential equations is a non-exact equation and solve by finding

an appropriate integrating factor.

1. 023

31 dxyxdyxxyx

2.

0)cos2()cos4sin2( 2 dyxydxxyxyy

3. 02

2sin4)2cos2(

dy

y

xxdxxy

4.

dyxyxdxxyxy )12()23( 323

5. dvuduvu )1()2(

6. 0cossincos2

2

dyxydxxxxy

7. 02

11 22

dyyexydxe xx

8. 0sin

2)cos(

dy

y

xxdxxy

9. 0)(2 dyxydxy

10. y (cosx-sin x) dx+cosx dy = 0

2.5.6 Homogeneous

In problems 1 – 3 solve the given homogeneous equation by using an appropriate substitution.

1. 02 22 dxydyyxx

2. 022 dxydyx

3. 02 yyxy

By Azizan & Faye, January 2014

4. 022 yyxxy

5. 022 dxyxdyxy

6. 022 dyyxydxxxy

In problems 4 – 7 solve the given IVP.

7. 10,022 yxydx

dyyx

8. 0)1(,012 ydyydxxyx

9. 11,0332 ydy

dxyxxy

10. 11,lnln yex

y

dy

dx

x

y

2.6.4 Bernoulli

Solve the given Bernoulli equation by using an appropriate substitution:

1. 4tyydt

dy

2. 2xexdt

dx t

3. txxtxdt

dxt 2

4. m

t

dt

dm

m

t

1

2

5. 01213 32 wxwdx

dwx

Solve the given Bernoulli equation subject to the respective initial conditions:

6. 032 32 dssds , 2

11

7. dtydyy 31 , 40 y

By Azizan & Faye, January 2014

8. ,22

2 tyy

dt

dy ,

10 y

9. xyydx

dy2cos2 , 10 y

10. yttydt

dy , 10 y

Final answers

2.1.4 Separable

(1) Cey x 3

3

1

(2) Cxxy cossin

(3)

4

2CxCxy

(4) Cey x 5ln

5

1

(5)1

)61(65

CCx

CxCy

(6)

)1(

)1(42

t

tx (7) )(

3

1tan 3 xy

(8)

42tan

2 xxy

(9)2

1cos2 xCey (10) 2627ln3

1 3 xy

2.2.6 Linear equations

(1)x

Cey 41

8

(2) Cx

xxxy

2

1cos (3)

2

3

53 t

Ctt (4) Cy csc

(5)t

t

te

C

t

ey

8

7

(6) tt

ttP cos1

sin)( (7)

1

ln)( R (8)3

2

2

1

tty

(9) 12

22

xe

yx

(10)

xxey .

2.3.5 Exact equations

(1) non-exact equation (2) Cu

uet t

2

3

2

22

(3) non-exact equation

(4) Cy

yxx 2

sincos2

(5) Cv

uvuvuf

2),(

2

(6) Cyxxyyxf 22),(

(7) non-exact equation (8) Cy

yxx 2

322

22

(9) Cf

322

2),(

(10) Ceyeyey yyy 2222

By Azizan & Faye, January 2014

2.4.3 Non-exact equations

(1) Cxyx

e y

3

3

(2) Cxyeye x

x

cos22

222

(3) Cxy

xy 2

2sin2 2

(4) Cy

xyx

223

(5) Cuu

v

1ln2

1

2 (6) Cx

ye x

cos2

22

(7) Ce

xexy

2

2

2

2

(8) Cxyxy sin2 (9) Cyxy 2/32/1

3

22

(10) Cxye x cos

2.5.6 Homogeneous equations

(1)

x

yxCy 22

(2) 1

Cx

xy (3) Cyx 22

(4) Cxy (5) Cxyx )2( 222

(6) Cyx 22 (7)

2

2

2 y

x

ey (8) 12 222 yxx (9) 12 333 xyx (10) yx

2.6.4 Bernoulli equations

(1) 3

1

3

3

1

tCety (2)

1

2

t

t

Cee

y (3)

11

1

tte

C

tx

(4) Ct

tm

ln (5) 3

1

3 2

1

xCew (6) 3

1

65

49

5

9

ss (7) 3

2

2

3

71t

ey

(8) 125

2

2

tey

t (9)

1

5

42cos

5

12sin

5

2

xexxy (10)

2

4

2

21

t

ey