Chapter 3 Exercises

By Azizan & Faye, January 2014

Chapter 3 (Higher order linear ODE)

3.1.9 Linear dependence/independence Determine whether the given sets of functions are linearly independent on the

interval ),( ∞−∞ .

1. 23

221 3)(,)(,2)( xxfxxfxxf ===

2. xxfxxfxf cos)(,)(,0)( 321 ===

3. xxfxxfxf 23

221 sin)(,cos)(,2)( ===

4. xxfxfxxf 2321 cos)(,1)(,cos)( ===

5. xxfxxf +=−= 1)(,1)( 21

6. xxx exfexfexf −− === )(,)(,)( 32

21

7. 1)( ,sin)(,2cos)( 32

21 === xfxxfxxf

8. 1)( ,)(,)( 321 === xfxxfexf x

9. xxfexfexf xx cos)( ,)(,)( 321 === −

10. xxfexfexf xx sin)( ,)(,)( 321 === −


3.2.4 Reduction of order

Find the second solution, if the first solution is given, as indicated.

1. 04 =+ʹ′ʹ′ yy ; xy 2sin1 =

2. 049 =+ʹ′ʹ′ yy ; xy 7cos1 =

3. 032 =−ʹ′−ʹ′ʹ′ yyy ; xey −=1

4. xeyyyy 31;096 ==+ʹ′−ʹ′ʹ′

5. xeyyyy x 3cos;074 21

−==+ʹ′+ʹ′ʹ′

6. xeyyyy ==+ʹ′−ʹ′ʹ′ 1;065

7. 01=ʹ′+ʹ′ʹ′ y

xy ; xy ln1 =

8. 0622 =−ʹ′+ʹ′ʹ′ yyy θθ ; 21 θ=y

9. xdtdx

dtxd 1242

2

=+ ; tex 21 =

10. xtdt

dxtdt

xd22

2 11=+ ; tx =1


3.3.5 Homogeneous linear with Constant Coefficients

For numbers 1 till 6, find the general solution of the given second-order

differential equation.

1. 096 =+ʹ′−ʹ′ʹ′ yyy

2. 023 =−ʹ′−ʹ′ʹ′ yyy

3. 025 =+ʹ′ʹ′ yy

4. 074 =+ʹ′+ʹ′ʹ′ yyy

5. 065 =+ʹ′−ʹ′ʹ′ yyy

6. 032 =−ʹ′+ʹ′ʹ′ yyy

For numbers 7 till 10, find the general solution of the third-order differential

equation.

7. 0=ʹ′−ʹ′ʹ′ʹ′ yy

8. 01025 =−ʹ′+ʹ′ʹ′−ʹ′ʹ′ʹ′ yyyy

9. 033 =−ʹ′+ʹ′ʹ′−ʹ′ʹ′ʹ′ yyyy

10. 08126 =−ʹ′+ʹ′ʹ′−ʹ′ʹ′ʹ′ yyyy


3.4.5 Non Homogeneous-Method Undetermined Coefficients

Solve the following Differential equation using the method of undetermined

coefficients with superposition principles.

1. xxyy +=−ʹ′ʹ′ 32

2. xeyy 2222 −+=ʹ′+ʹ′ʹ′

3. xxyy cos32sin4 +=+ʹ′ʹ′

4. xxexyyy −+=−ʹ′−ʹ′ʹ′ 22

5. xeyyy x cos52 =+ʹ′−ʹ′ʹ′

6. xxeyyy 2276 −−=−ʹ′−ʹ′ʹ′

7. )3cos(9 xxyy =+ʹ′ʹ′

8. xeyy 2252 +=ʹ′+ʹ′ʹ′

9. 231 xyy −=ʹ′ʹ′−ʹ′ʹ′ʹ′

10. )2cos(9 xxyy +=ʹ′+ʹ′ʹ′ʹ′


3.7.4 Non Homogeneous- Method of Variation of parameters

1. Solve each DE by variation of parameters method:

a. xyyy sin2 =−ʹ′−ʹ′ʹ′

b. xyy cos=ʹ′−ʹ′ʹ′

c. xxyy sincos=−ʹ′ʹ′

d. θθ=+ʹ′ʹ′ sinsecyy

e. αα=+ʹ′ʹ′ 2secsinyy

f. xexyy 4

1616 =−ʹ′ʹ′

g. 2

2

144

xeyyyx

+=+ʹ′−ʹ′ʹ′

h. tyy −=ʹ′+ʹ′ʹ′ 2

2. Solve by variation of parameters, subject to the initial conditions

.0)0(,1)0( =ʹ′= yy

a. 32

29

x

xeydxyd

=−

b. 132 2

2+=+− xy

dxdy

dxyd


3.8.5 Non homogeneous- Cauchy Euler

Solve the following differential equations.

1. xyyxyx ln2 =+ʹ′−ʹ′ʹ′

2. xxyyyx =++ʹ′+ʹ′ʹ′ 152

3. xx

yyx

y 212 =+ʹ′−ʹ′ʹ′

4. xexy

xyyx

−=+ʹ′−ʹ′ʹ′

322

5. xyxyx 292 =ʹ′−ʹ′ʹ′

6. yyxyx 442 −=ʹ′+ʹ′ʹ′

7. yx

yyx 414 −=ʹ′−ʹ′ʹ′

8. 42 =+ʹ′−ʹ′ʹ′ yyxyx

9. xxyyxyx sin64 42 =+ʹ′−ʹ′ʹ′

10. x

yyxyx 1332 =−ʹ′+ʹ′ʹ′


Final answers

3.1.9 Linear dependence/independence

(1) linearly dependent (2) linearly dependent (3) linearly dependent (4) linearly

dependent for πnx ±= , linearly independent for πnx ±≠

(5) linearly independent for 0≥x , linearly dependent for 0<x (6) linearly

independent

(7) linearly dependent (8) linearly independent (9) linearly dependent for

⎟⎠

⎞⎜⎝

⎛±=2πnx , linearly independent for ⎟

⎠

⎞⎜⎝

⎛±≠2πnx (10) linearly dependent for

πnx ±= , linearly independent for πnx ±≠

3.2.4 Reduction of Order

(1) xy 2cos2 = (2) xy 7sin2 = (3) xey 32 = (4) xxey 3

2 = (5)

xey x 3sin22

−=

(6) xey 51

2 = (7) 12 =y (8) 32

−=θy (9) tex 62

−= (10) t

x 12 =

3.3.5 Linear equations with constant coefficients

(1) xx xececy 32

31 += (2) xx

ececy 232

1 += (3) xcxcy 5sin5cos 21 +=

(4) ( )xcxcey x 3sin3cos 212 += − (5) x

x

ececy 25

1 += (6) xx

ececy 223

1 +=

(7) xx ececcy 321 ++= − (8) xcxcecy x 2sin2cos 325

1 ++=

(9) xcxccecy x sincos 3223

1 ++= (10) xxx excxececy 223

22

21 ++=

3.4.5 Method of Undetermined Coefficients

(1) y =C1ex +C2e

−x − 2x3 −13x ; (2) xx xexeCCy 2221

−− −++=

(3) xxxxCxCy cos2cos412sin2cos 21 +−+=


(4) y =C1e−x +C2e

2 x − x + 12− (16x2 + 1

9x)e−x

(5) xexeCxeCy xxx cos312sin2cos 21 ++=

(6) xxx exeCeCy 2721 )

910(

92 −− +−+=

(7) y =C1 cos3x +C2 sin3x +136

xcos3x + 112

x2 sin3x

(8) y =C1 +C2e−2 x +

52x + 14e2 x (9) y =C1 +C2x +C3e

x +52x2 + x3 + 1

4x4

(10) xxxCxCCy 2sin101

1813sin3cos 2

321 ++++=

3.7.4 Variation of parameters

(1a) xxeCeCy xx sin103

10cos

22

1 −++= − (1b) ( )xxeCCy x cossin21

21 +−+=

(1c) xeCeCy xx 2sin101

21 −+= −

(1d) [ ]θθθθθ tanseclncossincos 21 +−+= CCy

(1e) )sec(lnsincossincos 21 αααααα +++= CCy

(1f) xxxx exexeCeCy 42442

41 4

−−− −−+=

(1g) xxx exxxxeCeCy 21222

21 tan)1ln(

21

⎥⎦

⎤⎢⎣

⎡ −+−+= −

(1h) 2

32

21tteCCy t −++= −

(2a) 3/3/3/2

3/2

81

87

412xx

xx eexeexy −++−= (2b) xx eexy +−+= 2/44


3.8.5 Non-Homogeneous Cauchy Euler equation

(1) xxxCxCy ln2ln21 +++= (2) xxxxCxCy ln3

2

21 −++=

(3) xxCxCxxy lnln 212 ++= (4) xCxCxxey x

22

1)2( ++−= (5)

9210

21xxCCy −+= (6) 1)ln2sin()ln2cos( 21 ++= xCxCy (7)

xxxCxCy ln32

41 −+=

(8) 4ln21 ++= xxCxCy (9) xxxCxCy sin232

21 −+= (10)

xxCxCy41

23

1 −+= −

Chapter 3 Exercises

Documents

Transcript of Chapter 3 Exercises