Understanding Engineering Mathematics

(b) (i) Aex+Be−^2 x−

1

2

x−

3

4

(ii) (Ax+B)e^2 x+

1

4

x+

1

2

(iii) e−^2 x(Acosx+Bsinx)+

1

5

x+

1

25

(iv) Acos 2x+Bsin 2x+

1

4

x+

1

4

(v) Ae^3 x+Be−^3 x−

1

9

x−

1

9

(vi) Acosx+Bsinx+x+ 1

(c) (i) Aex+Be−^2 x−

1

3

xe−^2 x (ii) (Ax+B)e^2 x+

1

16

e−^2 x

(iii) e−^2 x(Acosx+Bsinx)+e−^2 x

(iv) Acos 2x+Bsin 2x+

1

8

e−^2 x

(v) Ae^3 x+Be−^3 x−

1

5

e−^2 x (vi) Acosx+Bsinx+

1

5

e−^2 x

(d) (i)Aex+Be−^2 x−

1

5

cosx−

3

5

sinx

(ii) (Ax+B)e^2 x+

8

25

cosx+

6

25

sinx

(iii) e−^2 x(Acosx+Bsinx)−

1

4

cosx+

1

4

sinx

(iv)Acos 2x+Bsin 2x+

2

3

sinx

(v)Ae^3 x+Be−^3 x−

1

5

sinx

(vi)Acosx+Bsinx−xcosx

8.(a) (i)

2

3

ex+

1

3

e−^2 x− 1

(ii)

(

x−

1

2

)

e^2 x+

1

2

(iii) −e−^2 x

(

2

5

cosx+

4

5

sinx

)

+

2

5

(iv)

1

2

( 1 −cos 2x)

(v)

1

9

(e^3 x+e−^3 x− 2 ) (vi) 2( 1 −cosx)

9.(a) (i)

e+ 1

e^2 +e+ 1

ex+

e^2

e^2 +e+ 1

e−^2 x− 1

(ii)

(

1

2

( 1 −e−^2 )x−

1

2

)

e^2 x+

1

2

(iii) e−^2 x

(

−

2

5

cosx+

2

5

(

cos 1−e^2

sin 1

)

sinx

)

+

2

5