Higher Engineering Mathematics

Assign-13-H8152.tex 23/6/2006 15: 13 Page 474

Differential equations

Assignment 13

This assignment 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 = 5

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

2. Determine the equation of the curve which satis-

fies the differential equation 2xy

dy

dx

=x^2 +1 and

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

3. A capacitorCis charged by applying a steady
   voltage E through a resistance R. The p.d.
   between the plates,V, is given by the differential
   equation:

CR

dV

dt

+V=E

(a) Solve the equation forEgiven that when time

t=0,V=0.

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

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

4. Show that the solution to the differential equa-

tion: 4x

dy

dx

=

x^2 +y^2

y

is of the form

3 y^2 =

√

x

(

1 −

√

x^3

)

given that y=0 when

x= 1 (12)

5. Show that the solution to the differential equation

xcosx

dy

dx

+(xsinx+cosx)y= 1

is given by:xy=sinx+kcosxwherekis 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=1 when

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 significant figures, in each of the two

numerical methods whenx= 1 .2. (30)

7. Use the Runge-Kutta method to solve the differ-

ential 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)