Example Sheet For Ordinary Differential Equations
Example Sheet For Ordinary Differential Equations
Example Sheet For Ordinary Differential Equations
Skills section
Questions in this section are intended to give practice in routine calculations. On this sheet
the skills questions start off with integration, as revision from Michaelmas Term, since
integration is an important mathematical operation in the solution of ordinary differential
equations.
Z
S1. Find the indefinite integral f (x) dx when f (x) is given by:
1
2+x
(1 + x)2
(a) (1 + x) 4
(b)
(c) 2x(3 + x2 )
(d) 2x sin(x2 )
(e) cot x
(h)
(i)
1
1 + x2
[Hint: substitution]
x
(1 x)(2 x)
(j) sin
1 x [Hint: substitution]
S2. Verify that the following are solutions of the corresponding differential equations
where in each case c, d, . . . are arbitrary constants.
dy
= x,
y = 21 x2 + c .
(a)
dx
dy
(b)
= y,
y = c ex .
dx
dy
d2 y
+5
+ 4y = 0 ,
y = c e4x + d ex .
(c)
2
dx
dx
dy
d2 y
+ 4y = 0 ,
y = cx4 + dx1 .
(d) x2 2 + 6x
dx
dx
d4 y
(e)
y = 0,
y = c sin x + d cos x + f ex + g ex .
dx4
2
8
dy
y = x2 .
+ 4y 3 = 6 ,
(f)
dx
x
S3. Solve by separation of variables:
(a)
dy
x3
=
.
dx
(y + 1)2
(b)
dy
4y
=
.
dx
x(y 3)
S4. Solve by the use of integrating factors:
(a)
dy
+ 2xy = 4x .
dx
(b)
dy
+ (2 3x2 )x3 y = 1 .
dx
Standard questions
5. Solve by change of variables and separation:
(x + y + 1)2
dy
+ (x + y + 1)2 + x3 = 0 .
dx
dy
+ (2x + 3y) = 0 .
dx
8. Solve:
(a)
y
dy
y
.
= + tan
dx
x
x
(b)
dy
y ln y = 0 .
dx
(Find a substitution that simplifies the equation.)
(ln y x)
(c)
xy
dy
+ (x2 + y 2 + x) = 0 .
dx
9. Solve the following differential equations subject to the initial conditions y(0) = 0,
y (0) = 1.
(a)
dy
d2 y
5
+ 6y = 0 .
2
dx
dx
(b)
d2
2
+n y = 0.
dx2
(c)
d2
d
+2
+4 y = 0.
dx2
dx
(d)
d2
+ 9 y = 18 .
dx2
(e)
d2
d
3
+ 2 y = e5x .
2
dx
dx
10*. A 20th century electrical circuit consists of a resistor (resistance R), an inductor
(inductance L), a capacitor (capacitance C) and a power supply (voltage V (t)) in
the arrangement shown in the diagram below. A current i(t) flows through the
resistor and a current j(t) flows through the inductor. There is therefore a current
i(t) + j(t) acting to increase the charge q(t) on the capacitor. The equations linking
the unknowns i(t), j(t) and q(t) are
iR = L
and
q
dj
= + V (t)
dt
C
dq
=i+j.
dt
i
R
j
L
q
i+j
V(t)
Second, consider the problem where V (t) = V0 sin(t) for t > 0, with initial conditions q = dq/dt = 0 at t = 0. Again solve for the evolution in each of the three
cases (a), (b) and (c) above. Show that at large times q(t) Q sin(t ) for
some constants Q and . Comment on the dependence of Q on , particularly
for large values of R.
11*. Find the complementary function, and a particular integral, of the equation
y (2 + c)y + (1 + c)y = e(1+2c)x
()
in the case c 6= 0.
(a) Show that there is a solution of the form y = f (x, c), where
f (x, c) = A ex + B
ex cx
ex
(e 1) + 2 (e2cx 2 ecx + 1) ,
c
2c
(b) Find the limit of f (x, c) as c 0. Hence find the complementary function and
a particular integral for () in the case c = 0.
12. Show that the equation
can be written as
d2 C
dC
=
2 d
d 2
d
ln
d
dC
(0)
d
dC
d
= .
2
C() = B + A
exp(t2 /4) dt .
dy
= ay + bx ,
dt
where a and b are constants, subject to the initial conditions x(0) = 2, y(0) = 1.
(b) Show that the pair of differential equations
dx
= x xy ,
dt
dy
= y + xy
dt
ex ey
x y
is independent of t.