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.
- Solve the differential equation:x
dy
dx
+x^2 =5given
thaty= 2 .5whenx=1. (4)
- Determine the equation of the curve which satisfies
thedifferential equation2xy
dy
dx
=x^2 +1andwhich
passes through the point (1, 2). (5)
- 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)
- 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)
- 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)
- (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)
- 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)