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 = 5 given
that y = 2.5 when x = 1.
(4)
2. Determine the equation of the curve which satisfies
the differential equation 2xy
d y
dx
= x 2 + 1 and which
passes through the point (1, 2).
(5)
3. A capacitor C is charged by applying a steady voltage E through a resistance R. The p.d. between the
plates, V , is given by the differential equation:
C R
dV
dt
+ V = E
(a) Solve the equation for V given that when time
t = 0, V = 0.
(b) Evaluate voltage V when E =50 V, C =10 μF,
R = 200 k and t = 1.2 s.
(14)
4. Show that the solution to the differential equation:
4x
d y
dx
=
x 2 + y 2
y
is of the form
3y 2 =
√
x
1 −
√
x 3
given that y = 0 when
x = 1.
(12)
5. Show that the solution to the differential equation
x cos x
d y
dx
+ (x sin x + cos x)y = 1
is given by: x y = sin x + k cos x where k is a
constant.
(11)
6. (a) Use Euler’s method to obtain a numerical
solution of the differential equation:
d y
dx
=
y
x
+ x
2
− 2
given the initial conditions that x = 1 when
y = 3, for the range x = 1.0 (0.1) 1.5
(b) Apply the Euler-Cauchy method to the differential 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 when x = 1.2
(30)
7. Use the Runge-Kutta method to solve the differential equation:
d y
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)