M1M1: Problem Sheet 1: Functions
1. Solve the following equations:
(a) x2 − 4x − 4 = 0
(c)
(b) 5x3 − 8x2 + 3x = 0
2u + 3
3u − 2
=
u−1
5u − 21
(d) 3 =
1
2
(
)
ev + e−v .
2.(a) If
x+1
x−1
and g(x) =
,
x−1
x+1
(except at the singular points) find expressions for f (g(x)), g(f (x)), f (f (x))
and g(g(x)). Give some thought to the points where f and g are “infinite.”
f (x) =
x−2
(b) Find the function inverse to f (x) = x−1
.
What does your result tell you about the graph y = f (x)?
3. More generally, consider functions f (x) of the form
f (x) =
ax + b
cx + d
d
for x ̸= − ,
c
(∗)
where a, b, c and d are real constants with c ̸= 0.
(a) Find the inverse function f −1 (x), assuming it exists. By computing
the composed function f −1 (f (x)), show that the inverse exists provided
ad ̸= bc. What happens in the special case ad = bc?
(b) Find the relation between a, b, c and d for which f −1 (x) ≡ f (x).
(c) Define a new function g(x) = f (x − e) where e = (a + d)/c. Use the
result of part (b) [remember it applies to ALL functions of the form
(*)] to show that g −1 (x) ≡ g(x).
4. The Heaviside step function H(x) is defined as follows:
{
1, x ≥ 0,
H(x) =
0, x < 0.
(i) Sketch graphs of the functions H(x − 1) and H(x) − H(x − 1);
(ii) The function f (x) is defined in terms of the functions a(x) and b(x) as
{
a(x), x ≥ 0,
f (x) =
b(x), x < 0.
Invent a single expression for f (x) in terms of a(x), b(x) and H(x).
5. We showed in lectures that any function defined over a suitable domain
can be written as the sum of an even and an odd function. Decompose the
following functions in this way, simplifying your answers where possible:
(
)1/2
1
1−x
; (b)
(a)
; (c) sin(x + 1).
x+1
1+x
6. The exponential function is defined by the infinite series
∞
∑
xn
f (x) ≡
n!
n=0
which is convergent for all values of x. By multiplying together the two series
expansions for f (x) and f (y) and collecting the first few terms of the same
degree (e.g. x3 , x2 y, xy 2 and y 3 are all of degree 3), verify that
f (x)f (y) = f (x + y).
(1)
[To attempt a full proof, look at Progress Test 1 from 2006, on the website.]
7. In lectures we showed the exponential function y = f (x) defined in Q6 is
strictly increasing, and so has an inverse function, y = g(x). Only using the
property (1) and general results about inverses, establish the results:
( )
(u)
1
g(uv) = g(u) + g(v); g
= g(u) − g(v).
= −g(u); g
u
v
8. Find all solutions x to the following two equations:
(a) 4 sin2 x − 5 cos x − 5 = 0;
(b) 2 sec2 x − tan x − 3 = 0.
9. Use the identities
sin(x + y) = sin x cos y + sin y cos x,
cos(x + y) = cos x cos y − sin x sin y,
to derive expressions for sin 2θ, cos 2θ, tan 2θ, sin α + sin β, cos α + cos β.
And remember them.
10. Given that t = tan x2 , show that
sin x =
1 − t2
2t
and
cos
x
=
.
1 + t2
1 + t2
2
M1M1: Problem Sheet 2: Power series expansions
1. In lectures (http://www.ma.ic.ac.uk/∼ajm8/M1M1/cozzin.pdf)
we proved that for cos and sin defined by infinite series,
(a) cos(x + y) = cos x cos y − sin x sin y
(b) there is a number τ with 1.4 < τ < 1.6 such that cos τ = 0.
Using only these results and others derived from the infinite series, show that
(i)
cos2 θ+sin2 θ = 1,
(iv)
cos 4τ = 1
(ii)
cos(θ+τ ) = − sin θ
(iii)
cos(2τ ) = −1
(v)
cos(θ + 4τ ) = cos θ
for all θ.
2. The hyperbolic sine and cosine, denoted sinh x and cosh x, are respectively
the odd and even parts of the exponential function exp(x). Use this fact to
write down the series expansions for sinh x and cosh x.
The hyperbolic tangent, tanh x, is defined as
tanh x =
sinh x
.
cosh x
Use the series expansions for sinh and cosh to find the first 3 non-zero terms
in the series expansion of tanh x.
3. Derive the following expression for the inverse hyperbolic tangent
1
1+x
−1
tanh x = log
.
2
1−x
Use this expression to find the series expansion of tanh−1 x.
4. Using well-known series expansions, but without using Maclaurin ideas,
find the first three non-zero terms in the power series of:
(a) (1 + x)exp(x); (b) sin(x + 1); (c) exp(x) log(1 + x);
1
(d)
; (e) sec x; (f ) tan x;
2 − exp(x)
(g) log(1 + exp(x)); (h) cos(sin(x)).
[Remember – only expand in variables which are small as x → 0]
5. Find the Maclaurin series for the function
f (x) =
x2
1
.
+ 3x + 2
[Hint - how can you render the repeated differentiation easier to perform?]
6. Find the first three non-zero terms of the Maclaurin series of the functions
(a) exp(x) cos x;
(b) tan−1 (x);
(c) sec x.
Have you any ideas how you might find ALL terms in the series for (a) and
(b) more easily?
Check that your result for (c) agrees with Q4.
M1M1: Problem Sheet 3: Convergence of Power Series and Limits
1. Use the Ratio Test to find the Radii of Convergence of the power series
for
(a) cos(x)
(b) log(1 + x)
(c) (1 + x)α
where α is not a positive integer.
2. Consider the two functions
f (x) =
1
2 − cosh x
g(x) =
1
.
1 + exp(x)
(a) Explain why it is to be expected√that the Radius of Convergence of the
Maclaurin series for f (x) is log(2 + 3).
(b) It is found using a computer that the power series for g(x) appears to
have a radius of convergence R where 3.1 < R < 3.2. Can you think of a
reason why this might be? [Hint – think of x as a complex number].
3. Evaluate the following limits:
x3 − 1
x2 + 5x + 6
x5 + 7x3
(a) lim
; (b) lim 2
; (c) lim
x→1
x→−2 x + 3x + 2
x→∞ 4x5 + x2
x
3
2 1/2
x −1
(1 + x )
1 + cos x
(d) lim
; (e) lim
; (f ) lim
;
x→1 x − 1
x→∞
x→π tan2 x
x
sin(x − 1)
tan(p(x − 2))
tan x
; (h) lim 2
; (i) lim
(g) lim
x→1 x − 5x + 4
x→2 tan(q(x − 2))
x→0
x
4. Use the result that lim xe−αx = 0 for α > 0 to show that
x→∞
lim+ tα log t = 0
t→0
where the notation t → 0+ means that t tends to zero through positive values.
5. Evaluate the following limits:
log x
x + sin x
1 − sin x
; (b) lim 2
; (c) lim
;
2
x→1
x→π/2 (x − π/2)2
x+x
x −1
h
i
x−2
(d) lim (sec x−tan x); (e) lim (sec x)
; (f ) lim x1/3 (x + 1)2/3 − x2/3 ;
(a) lim
x→0
x→0
x→π/2
(g) lim
x→0
1
1
−
x sin x
x→∞
c x
x2 − 2x + 1
; (i) lim 4
; (h) lim 1 +
x→1 x + x3 − 7x + 5
x→∞
x
6. For each positive integer n, we define a function
n
if
1/n < x < 2/n
fn (x) =
0 otherwise
What is the maximum value of fn (x), and what is its integral over all x?
Show that for each value of x, lim fn (x) = 0. Deduce that
n→∞
lim
n→∞
 ∞

Z∞
Z
h
i
max[fn (x)] 6= max lim (fn (x)) and
lim [fn (x)] dx 6= lim  fn (x) dx
x
x
n→∞
n→∞
−∞
n→∞
−∞
M1M1: Problem Sheet 4:
Differentiation
1. Find the derivatives of the following function from first principles (i.e.,
use the definition of the derivative and algebraic manipulation; strictly, you
should not use Taylor series or the binomial series for non-integers):
(a) x3 ;
(b) x1/3 ;
(c)
√
x2 − 1;
(d) cos x;
(e) tan x;
(f ) sin
√
x.
2. Using any rules of differentiation that you like (e.g. the product rule,
quotient rule, chain rule), find the derivatives of the following functions:
(a) sin(x2 ); (b) sin2 x; (c) sin(2x); (d) 10x ; (e) (sin x)x ; (f ) tan(sin x);
(g) cot−1 (x); (h) exp(3x2 +5x+2); (i) exp(−x)cosh(2x); (j) log(x)exp(x);
(k) sin−1 (x); (l) sec x; (m) log | sec x + tan x|; (n) (x2 − 1)1/2 ;
1
x
; (p) 2
; (q) log(x) sin−1 (x); (r) tan−1 (x).
(o) 2
1/2
(x − 1)
(x − 1)3/2
3. A curve in the (x, y)-plane is given in polar coordinates (r, θ) (where
x = r cos θ, y = r sin θ) by the equation r(θ) = sin θ. Show that
dy
= tan 2θ.
dx
Another curve is given by the equation r(θ) = 1+sin2 θ. Find
corresponding to θ = π/4.
dy
dx
at the point
4. Find the location and nature of the stationary points of the following
curves:
(a) y = 2x3 + 15x2 − 84x;
(b) y = x5 − 5x + 1;
1
(c) y = log 2x + .
x
5. If
x(s) = cos 2s,
show that
dy
1
=
dx
2
y(s) = s − tan s,
s
1−x
.
(1 + x)3
6. Given that xy(x + y)a = b, where a and b are constants, show that
µ
¶
y (a + 1)x + y
dy
.
=−
dx
x (a + 1)y + x
7. Establish the relation
d2 y
d2 x
=
−
dy 2
dx2
µ
dy
dx
¶−3
.
Verify this expression for the example y = (1 + x)−1 .
8. If
h
y = x+1+
show that
√
√
x2
x2 + 2x + 2
ip
+ 2x + 2 ,
dy
= py.
dx
Differentiate this equation to show that
(x2 + 2x + 2)
dy
d2 y
+ (x + 1) − p2 y = 0.
2
dx
dx
Now use Leibniz’s formula to show that, at x = 0,
n
dn+2 y
dn+1 y
2
2 d y
2 n+2 + (2n + 1) n+1 + (n − p ) n = 0.
dx
dx
dx
9. Use Leibniz’s formula to show that:
¤
dn £
2
(1
+
x
)exp(x)
= 1 + n(n − 1) when x = 0.
dxn
10. Find f (n) (x) if
f (x) =
x2
1
.
+ 3x + 2
11. A sheet of metal of fixed area A is to be made into a right circular cylinder
with closed ends. Show that the volume of the cylinder is a maximum when
its length is equal to its diameter and find this maximum volume.
12. Prof Liegroup keeps a clue to the Treasure Hunt on the top of a high
shelf of his office, only accessible with a ladder. To reach his office, you have
to walk down two passages, of widths a and b, which meet at a right-angled
corner. Show that the longest ladder that can be carried horizontally around
this corner is of length (a2/3 + b2/3 )3/2 .
2
M1M1: Problem Sheet 5:
Curve sketching
1. Sketch the graph of the function
y=
x+3
,
2x + 1
carefully indicating any special features on your graph.
2. Sketch graphs of the functions:
(a) x2 − 6x + 8; (b) x exp(−x2 );
3. If
(c)
cos x + sin x
√
.
2
x2 + 2x + 5
,
x+1
y(x) =
show that
4
.
x+1
Hence sketch the graph of the function y = y(x), carefully identifying any
asymptotes, stationary points or points of inflexion.
y(x) = x + 1 +
4. Sketch the graph of the function
y=
x(x − 2)
,
x−3
carefully indicating any special features on your graph.
5. Sketch a graph of the function
y 2 = x(x2 − 1),
carefully indicating any special features on your graph.
6. A curve is given, in polar coordinates, by the formula
r=
Sketch this curve.
2
.
3 + cos θ
7. A curve is given in parametric form as
x(t) = a cos3 t; y(t) = b sin3 t,
where a and b are positive constants. Sketch the curve carefully, noting how
dy/dx behaves at the maximum and minimum values of y.
8. Consider the function
x3 − 1
f (x) = 3
.
x +1
(a) Put f (x) in partial fraction form;
(b) Find and classify all the stationary points of f (x);
(c) Find all the points of inflexion;
(d) Sketch the graph of f (x) carefully indicating all the important features
of the graph on your sketch (including the stationary points and points
of inflexion).
9. Plot the curve given parametrically by
x = cos t,
y=
| sin t|
t
for −2π ≤ t ≤ 2π.
[Hint: Consider carefully what happens as t passes through zero.)
10. The function h(x) is defined for x > 1/2 by
q
q
√
√
h(x) = x + 2x − 1 − x − 2x − 1
Sketch the curve y = h(x) for x > 1/2.
[Important hint: before you start, try to simplify h(x), by considering h2 .
Recall that the square roots always take positive values.]
2
M1M1: Sheet 6: Mean Value Theorem and Taylor’s Theorem
1. Consider the function f (x) = x3 − 8x − 5 in the domain −1 ≤ x ≤ 4.
Find a number c, with −1 < c < 4, such that
f (4) − f (−1)
= f 0 (c).
5
Now let f (x) = x4 . Show that there is no number c with −1 < c < 4 such
that
f (4) − f (−1)
= f 0 (c).
5
Why does this not contradict the mean value theorem?
2. Apply the mean value theorem to tan−1 (x) to show that
b−a
b−a
< tan−1 (b) − tan−1 (a) <
2
1+b
1 + a2
where 0 < a < b. Hence obtain the value of tan−1 (21/20) to 2 decimal places
of accuracy.
3. Use the mean value theorem to show that
√
1
1
√ < 66 − 8 < .
8
66
4. Use the first few terms of an appropriate series expansion to estimate the
value of 8.11/3 , giving your answer correct to four decimal places.
5. Find the first few terms in the Taylor series of sin(x + a) about x = 0 and
hence verify the identity
sin(x + a) = sin x cos a + sin a cos x
6. Let
y(x) = sin(m sin−1 (x))
where m is some real number. Show that
(1 − x2 )y 00 − xy 0 + m2 y = 0
By differentiating this equation n times, show that
n
dn+2 y
dn+1 y
2
2 d y
−
(2n
+
1)x
+
(m
−
n
)
= 0.
dxn+2
dxn+1
dxn
Now set x = 0 and hence derive the following Taylor expansion about x = 0:
(1 − x2 )
x3
x5
+ m(1 − m2 )(9 − m2 ) + . . .
3!
5!
Show that this series converges for |x| < 1.
y(x) = mx + m(1 − m2 )
7. Show that y(x) = tan(x) satisfies the equation
dy
= 1 + y2.
dx
By repeated differentiation of this equation, find the higher derivatives of y(x)
and hence determine the first three non-zero terms of the Taylor expansion
of tan x about x = 0. Check your answer by using the series for sin x/ cos x.
8. Derive the Taylor series about x = 0 for the function
1+x
log
.
1−x
State its radius of convergence and use the series to obtain the value of
log(5/3) to 4 decimal places of accuracy.
9. Let f (x) be differentiable in the neighbourhood of x = a as many times
as we like. Use an infinite Taylor series to show that, when h is small,
f (a + h) − f (a − h)
f 0 (a) =
+ E1
2h
where the error
h2 f 000 (a)
E1 = −
+ O(h4 ).
6
Show also that
f (a + h) − 2f (a) + f (a − h)
f 00 (a) =
+ E2
h2
where the error
h2 f 0000 (a)
E2 = −
+ O(h4 ).
12
Now let f (x) = sin x and h = π/12. From the above approximations, find
the values of f 0 (π/4) and f 00 (π/4) and compare with the exact values.
2
M1M1: Problem Sheet 7: Integration
1. Integrate the following functions of x:
(a)
x+1
;
x
x
;
x+1
(b)
(c)
x+1
.
x−1
2. Using partial fractions (or otherwise) find the integrals of
(a)
1
;
x(2 − 3x)
(b)
x
;
2
x −1
(c)
x2
;
x3 − 1
(d)
1
.
+ 1)
x(x2
3. Use an appropriate substitution to integrate:
(a)
ex
;
ex + 1
(b) sin2 x cos x;
(e) √
1
x2
−1
(c)
;
sin x
;
1 + cos x
(d) √
1
;
1 − x2
(f ) cos x esin x .
4. Use ‘integration by parts’ to integrate:
(a) ex cos x;
(b) log x;
(c) x2 cos x;
(d) cos−1 x.
5. Integrate the following functions (using any appropriate method):
√
√
x
tan−1 (x)
2
; (c) cosec x; (d)
(a) x − 1; (b)
.
1+x
1 + x2
Evaluate the following definite integrals:
Z π/2
dx
dx
; (f )
;
(e)
5 + 4 cos x
(sin x + cos x)2
0
0
Z 2
Z 1
dx
dx
; (h)
.
(g)
2
3/2
2
0 (1 + x )
1 x + 3x + 1
Z
π/2
6. Show that [NB MAPLE can’t do this!]
Z
Z π
x dx
π π
dy
π2
.
=
=
2
2 0 1 + cos2 y
23/2
0 1 + cos x
7. Show that
Z
π
0
dx
π
,
=√
2
α − cos x
α −1
α > 1.
8. The integrals I(t0 ) and J(t0 ) are defined as
Z ∞
dt
I(t0 ) =
, t0 > 0,
−∞ cosh t + cosh t0
Z ∞
dt
J(t0 ) =
, 0 < t0 < π/2.
−∞ cosh t + cos t0
Using the substitution p = et , or otherwise, show that
I(t0 ) =
9.
2t0
,
sinh t0
J(t0 ) =
2t0
.
sin t0
Determine whether or not the integral
Z ∞
f (x) dx
exists in the following cases:
0
√
(a) f (x) = xα 2 + cos x
where α is a real constant.
(b) f (x) = (x2 − 1)−1/3 .
(c) f (x) =
1
.
(x + 1) log x
(d) f (x) =
(x + 1) sin x
.
x3/2 (x − π)
10. If
Fn =
show that
Z
1
xn ex dx
0
Fn = e − nFn−1 ,
n = 1, 2, ...
Hence, evaluate F4 .
11. If
In =
Z
∞
2
xn e−x dx
0
where n is a positive integer, show that
n−1
In−2
In =
2
Hence, evaluate I5 .
2
for n > 1.
12. Given that
un (x) =
Z
n
x cos x dx,
vn (x) =
Z
xn sin x dx,
by performing a single (complex) integration, show that
un (x) = xn sin x − nvn−1 (x)
vn (x) = −xn cos x + nun−1 (x)
Hence, evaluate
Z
13. If
In =
show that for n > 1
In =
Hence, evaluate I5 .
14. If
In =
Z
1
x4 sin xdx.
Z
π/4
tann xdx,
0
1
− In−2 .
n−1
√
xn 1 + x dx, n = 0, 1, 2, . . .
0
show, by first bounding the integrand, that
√
1
2
< In <
.
n+1
n+1
Derive the recurrence relation for n > 0
(3 + 2n)In = 25/2 − 2nIn−1 .
From the two previous results, deduce that
√
2
In >
,
n + 3/2
and using the various inequalities you have found, prove that as n → ∞,
√
nIn → 2.
3
Line, Area and Volume integrals
15. Find the length of the curve y =
√
1 − x2 between x = 0 and x = 1.
16. The cycloid curve we considered in lectures was given parametrically by
x = t − sin t, y = 1 − cos t. Find the length of this curve between the points
(0, 0) and (2π, 0).
17. The portion of the parabola y = 1 − x2 in y > 0 is rotated about the
x-axis. What is the volume of the shape so formed?
18. The curve y = 1/x for x > 1 is rotated about the x-axis. Calculate,
if possible, the volume and the surface area so produced. Do you find the
result strange?
19. Consider the integral of exp(−x2 − y 2 ) over the whole (x, y)-plane,
ZZ
2
2
e−x −y dA.
I=
R2
(a) Using polar coordinates, and performing the r and θ integrals separately,
show that I = π.
(b) using Cartesian coordinates, show that, I = J 2 where
Z ∞
√
2
J=
e−t dt,
and deduce that J = π.
−∞
20. Show that the area of the ellipse
x2 y 2
+ 2 =1
a2
b
is πab.
Choosing a suitable parameterisation, show that its circumference, L, is
Z π
Z 2π
1/2
2
2
2
2
a sin θ + b cos θ
dθ = C
(1 − k cos φ)1/2 dφ,
L=
0
0
for some constants C and k. (N.B. This integral cannot be evaluated in terms
of elementary functions. It is known as an elliptic integral.)
This is quite a long sheet. If you don’t manage to do it all, try at least a
representative mixture from questions 1-8, 9, 10-14 and 15-20.
4
M1M1: Problem Sheet 8: Complex numbers
1. Put the following complex numbers into standard form, i.e., in the form
x + iy for some real x and y:
(a)
1
;
2 + 3i
(f )
(b)
1
;
2 − 3i
(c)
(1 + i)(2 + i)(3 + i)
;
(1 − i)
1−i
;
1+i
(g)
(d)
1
√ ;
1 + 3i
1+i
;
1−i
(h)
(e)
√
1
;
i5
5 + 12i.
2. Let z1 = −1 + 2i and z2 = 3 − 2i. Find the standard form of the following
complex numbers:
(a) 2z1 − 3z2 ;
(b) z1 z2 ;
(c)
z12
;
z2
(d) |z12 z2 |.
Now let z1 = 1 − 3i and z2 = 3 − 2i. Find the standard form of the following
complex numbers:
z1 (a) |z1 + z2 |; (b) |z2 |z1 ; (c) z1 + |z1 |; (d) .
z2
3. Write the following complex numbers in standard form:
(a) 2eiπ/2 ;
(b) 3e−iπ ;
(c) 2e−iπ/2 ; (d) 3eiπ/4 ; (e) 2eiπ/6 .
Write the following complex numbers in polar form:
√
1+i
(a) i; (b) √ ; (c) − 1 + i 3; (d) 6 + 8i;
2
(e) − 1.
4. By expressing −1+i in polar form (i.e., in the form reiθ ), find the standard
form of the number (−1 + i)−8 .
5. Show that
1 + cos θ + i sin θ
= cot
1 − cos θ + i sin θ
θ i(θ−π/2)
.
e
2
6. Given that 2 + i is a solution of the equation
z 4 − 2z 3 − z 2 + 2z + 10 = 0,
find the other solutions.
7. If z = eiθ , show that
1
cos(nθ) =
2
1
z + n
z
n
.
Hence, show that
1
3
15
5
cos(6θ) +
cos(4θ) +
cos(2θ) + .
32
16
32
16
cos6 θ =
8. Use the fact that
cos(nθ) = Re[einθ ]
to establish the identity
1 + cos(θ) +
cos(2θ) cos(3θ) cos(4θ)
+
+
+ ∙ ∙ ∙ = ecos(θ) cos(sin(θ)).
2!
3!
4!
9. Sketch the following curves in the complex plane:
(b) |z − i| = 2; (c) Re[z 2 ] = 1;
π
z+1
= ± , z 6= ±1
(e) arg
z−1
2
(a) |z − i| = |z − 1|;
(d) zz = 1;
10. By treating it as a quadratic, find the roots of the equation
z 2i + z i + 1 = 0.
11. Three roots of a polynomial equation of degree 5 with real coefficients
are 1, i ± 1. Find the equation.
12. Find all the values of log(1 + i) and log (1 + i)1/i .
13. Find all complex solutions to the following equations:
√
(a) ez = −2; (b) z 7 = −1; (c) cos z = 2.
13 12 . Infer the radius of convergence of the Maclaurin series of the function
1/(2 + ex ).
14. Use de Moivre’s theorem to show that
cos(5θ) = 16 cos5 (θ) − 20 cos3 (θ) + 5 cos(θ),
sin(5θ) = sin θ 16 cos4 (θ) − 12 cos2 (θ) + 1 .
15. Use the formula:
1 − z n+1
1 + z + z + z + z + ∙∙∙ + z =
1−z
2
3
4
n
to prove Lagrange’s identity:
N
X
n=0
cos(nθ) =
1 sin[(N + 1/2)θ]
+
.
2
2 sin(θ/2)
2
M1M1: Problem Sheet 9
Ordinary Differential Equations
1. Find the general solution of the 2nd order ODE
y 00 + my 0 + 2y = 0
√
in the 3 cases (a) m = 3 (b) m = 2 and (c) m = 2 2.
2. Find the solution to the problem for general m 6= 1
y 00 + (m + 1)y 0 + my = 0,
satisfying y(0) = 0, y 0 (0) = 1.
Take the limit of y as m → 1, and verify that what you obtain does indeed
solve the problem when m = 1.
1. Solve the following differential equations:
dy
2x
=
;
dx
(y + 1)
dy
(x + y)
=
;
(d)
dx
(x − 2y)
dy
(g) x
= y + xey/x ;
dx
(a)
dy
dy
(1 + y)
= (1 + x)(1 + y);
(c)
=
;
dx
dx
(2 + x)
dy
dy
x3 + y 3
(e) xy
= x2 + y 2 ;
=
(f ) y 2
;
dx
dx
x
dy
dy
4 log x
2
2
(h) xy
;
= x2 e−y /x + y 2 ; (i)
=
dx
dx
y2
(b)
2. By making a substitution of the form y = at + bx + c, solve the following
differential equations for x(t):
(a)
dx
t−x+2
=
;
dt
t−x+3
(b)
dx
1 − 2x − t
=
dt
4x + 2t
3. Find the solutions of the following initial value problems:
dx
dx
− 2t(2x − 1) = 0, x(0) = 0; (b)
+ 5x − t = e−2t , x(−1) = 0;
dt
dt
dx
dx
+ x cot t = cos t, x(0) = 0; (d) (1 + t2 ) + 3xt = 5t(1 + t2 ), x(1) = 2;
(c)
dt
dt
(a)
4. Solve
dy
2
.
=
dx
x + ey
5. By using a suitable substitution (or otherwise), find the solution of
y(xy + 1) + x(1 + xy + x2 y 2 )
dy
= 0.
dx
6. Solve
r tan θ
a2 − r 2
dr
dθ
= 1,
r
π 4
= 0.
7. Find the general solution R(t) of
d2 R 2 dR
−
= t4 .
dt2
t dt
8. Find x(t) and y(t) satisfying the coupled system of first-order differential
equations given by
dy x
+ = 1,
dt
y
dx
dy
−x
= 2ty 2 .
y
dt
dt
with x(0) = 0 and y(0) = 1.
9. (Difficult). Find the solution R(t) of the nonlinear second-order equation
d2 R 1
1=R 2 +
dt
2
satisfying the conditions R(0) = 1 and
Hint: try finding
dR
dt
dR
dt
dR
(0)
dt
2
.
= 0.
as a function of R(t).
10. Solve the equation
dy
+ sin
dx
dy
dx
=x
2
with y(0) = 0.