Understanding Engineering Mathematics

Answers

1. (i)

1

4

(ex−e−^3 x) (ii) −

1

4

cos 2x−

π

16

sin 2x+

1

4

x+

1

4

(iii)

2

3

sinx−

1

3

sin 2x

1

6

e^3 x−

3

2

ex+x+

4

3

15.7 Reinforcement

In all these exercises check your results by substituting back to the equation – the practice
will do you good!

1.Radioactive material decays at a rate proportional to the amount present. Construct
and solve a mathematical model giving the amount of material remaining after a
given time.

2.Solve the following differential equations subject to the conditions given:

(i) y′=xy( 0 )= 1

(ii) y′=cosxy(π)= 0

(iii) xy′=x^2 + 1 y( 1 )= 0

(iv) y′′= 4 y( 0 )= 1 y′( 0 )= 2

(v) y′′=x^2 − 1 y( 0 )= 0 y( 2 )= 1

(vi) y′′=cosxy( 0 )= 0 y(π)= 1

(vii) y′= 3 y^2 y( 0 )= 1

(viii) y′=secyy( 0 )=π

3.Find the general solution of the differential equationsy′=f(x,y)wheref(x,y)is
given by:

(i) xy^2 (ii)

y

x

(iii) xsecy

(iv) ex+y (v) 10− 2 y (vi) e^2 x−^3 y

(vii)

2 x

5 −siny

(viii) ylnx (ix) (x−y)/x

(x) y(x+ 2 y)/[x( 2 x+y)](xi)3(y^2 − 3 y+ 2 )

4.Solve the equations:

(i) y′=e^2 x−y (ii) y′+ 2 y= 3 ex (iii) y′+xy=x^3

(iv) xy′= 3 y− 2 x (v) (x^2 − 1 )y′+ 2 y= 0

(vi) (x− 1 )y′= 3 x^2 −y (vii) xy′− 2 y=x^3 e−^2 x