Understanding Engineering Mathematics

15.9 Answers to reinforcement exercises

1. m=moe−λt


2. (i)

x^2

2

+ 1 (ii) sinx (iii)

1

2

(x^21 )+lnx

(iv) 2x^2 + 2 x+1(v)

x^4

12

−

x^2

2

+

5

6

x

(vi) −cosx−

1

π

x+ 1 (vii)

1

1 − 3 x

(viii) sin−^1 x

3. (i) −

2

x^2 +C

(ii) Cx (iii) sin−^1

(

x^2

2

+C

)

(iv) −ln|C−ex| (v)

1

2

( 10 −Ce−^2 x) (vi)

1

3

ln|C+

3

2

e^2 x|

(vii) 5y+cosy=x^2 +C (viii) Cxxe−x (ix)

x

2

( 1 −Cx)

(x) (y−x)^3 =Cx^2 y^2 (xi)

y− 2

y− 1

=Ce^3 x

4. (i) Ce−x+

1

3

e^2 x (ii) ex+Ce−^2 x

(iii) x^2 − 2 +Cexp(−x^2 / 2 ) (iv) x+Cx^3

(v) C

(

x− 1

x+ 1

)

(vi)

x^3 +C

x− 1

(vii) Cx^2 −

x^2

2

e−^2 x

5. (i)

1

2

(e^5 x−e^3 x) (ii) x^2 (lnx+ 2 )

(iii)

1

4

(

x^2 −

1

x^2

)

(iv) −

1

x^3

(cosx+ 1 )

6. (i) Aex+Be−^2 x (ii) (Ax+B)e^2 x

(iii) e−^2 x(Acosx+Bsinx) (iv) Acos 2x+Bsin 2x

(v) Ae^3 x+Be−^3 x (vi) Acosx+Bsinx

7.(a) (i) Aex+Be−^2 x− 1 (ii) (Ax+B)e^2 x+

1

2

(iii) e−^2 x(Acosx+Bsinx)+

2

5

(iv) Acos 2x+Bsin 2x+

1

2

(v) Ae^3 x+Be−^3 x−

2

9

(vi) Acosx+Bsinx+ 2