Higher Engineering Mathematics, Sixth Edition

Revision Test 14

This Revision Test covers the material contained in Chapters 46 to 49.The marks for each question are shown in
brackets at the end of each question.

1. Solve the differential equation:x

dy

dx

+x^2 =5given

thaty= 2 .5whenx=1. (4)

2. Determine the equation of the curve which satisfies

thedifferential equation2xy

dy

dx

=x^2 +1andwhich

passes through the point (1, 2). (5)

3. A capacitorCis charged by applying a steady volt-
   ageEthrough a resistanceR. The p.d. between the
   plates,V, is given by the differential equation:

CR

dV

dt

+V=E

(a) Solve the equation forVgiven that when time

t=0,V=0.

(b) Evaluate voltage V when E= 50 V,C= 10 μF,

R=200kandt= 1 .2s. (14)

4. Show that the solution to the differential equation:

4 x

dy

dx

=

x^2 +y^2

y

is of the form

3 y^2 =

√

x

(

1 −

√

x^3

)

given that y=0when

x= 1. (12)

5. Show that the solution to the differential equation

xcosx

dy

dx

+(xsinx+cosx)y= 1

is given by: xy=sinx+kcosx where k is a

constant. (11)

6. (a) Use Euler’s method to obtain a numerical
   solution of the differential equation:
   dy
   dx

=

y

x

+x^2 − 2

given the initial conditions thatx=1when

y=3, for the rangex= 1 .0 (0.1) 1.5

(b) Apply the Euler-Cauchy method to the differ-

ential equation given in part (a) over the same

range.

(c) Apply the integrating factor method to

solve the differential equation in part (a)

analytically.

(d) Determine the percentage error, correct to 3 sig-

nificant figures, in each of the two numerical

methods whenx= 1. 2 (30)

7. Use the Runge-Kutta method to solve the dif-
   ferential equation:

dy

dx

=

y

x

+x^2 −2 in the range

1.0(0.1)1.5, given the initial conditions that at

x=1,y=3. Work to an accuracy of 6 decimal

places. (24)