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Mathematical Tripos Part IA

C. P. Caulfield

Differential Equations A3

Michaelmas 2013

DIFFERENTIAL EQUATIONS

Examples Sheet 1

The starred questions are intended as extras: do them if you have time, but not at the expense of

unstarred questions on later sheets.

1. Show, from first principles, that, for non-negative integern d

dxxn =nxn1.

2. Let f(x) =u(x)v(x). Use the definition of the derivative of a function to show that

df

dx=u

dv

dx+

du

dxv.

3. Calculate

(i) d

dx

ex

2

sin2x

,

(ii) d12

dx12(x cos x) using (a) the Leibniz rule and (b) repeated application of the product rule,

(iii) d5

dx5

[ln(x)]2

.

4. (i) Write down or determine the Taylor series forf(x) =eax about x= 1.

(ii) Write down or determine the Taylor series for ln(1 +x) about x = 0. Then

show that

limk

k ln(1 +x/k) =x

and deduce that

limk

(1 +x/k)k =ex.

What property of the exponential function did you need?

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5. Determine by any method the first three non-zero terms of the Taylor expansions aboutx= 0of

(i) (x2 +a)3/2,

(ii) ln(cos x),

iii) exp 1

(x a)2

,

where ais a constant.

6. By considering the area under the curvesy= ln xand y = ln(x 1), show that

Nln NN

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*9. In a large population, the proportion with income betweenxand x+ dxis f(x)dx. Expressthe mean (average) income as an integral, assuming that any positive income is possible.

Let p= F(x) be the proportion of the population with income less than x, and G(x) be themean (average) income earned by people with income less than x. Further, let (p) be the

proportion of the total income which is earned by people with income less than xas a functionof the proportion p of the population which has income less than x. ExpressF(x) and G(x) asintegrals and thence derive an expression for (p), showing that

(0) = 0, (1) = 1

and

(p) =F1(p)

, (p) =

1

f(F1(p))>0.

Sketch the graph of a function (p) with these properties and deduce that unless there iscomplete equality of income distribution, the bottom (in terms of income) 100p% of thepopulation receive less than 100p% of the total income, for all positive values ofp.

10. For f(x, y) = exp(xy), find (f/x)y and (f/y)x. Check that 2f

xy =

2f

yx. Find

(f/r) and (f/)r,

(i) using the chain rule,

(ii) by first expressing fin terms of the polar coordinates r, ,

and check that the two methods give the same results.

[Recall: x= r cos , y=r sin . ]

11. If xyz+x3 +y4 +z5 = 0 (an implicit equation for any of the variablesx, y,zin terms of theother two), find

x

y

z

,

y

z

x

,

z

x

y

and show that their product is1.Does this result hold for an arbitrary relation f(x,y,z) = 0 ?

What about f(x1, x2, , xn) = 0 ?

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12. In thermodynamics, the pressure of a system,p, can be considered as a function of thevariablesV(volume) andT(temperature) or as a function of the variablesV and S(entropy).

(i) By expressing p(V, S) in the form p(V, S(V, T)) evaluate

pV

T

p

VS

in terms of S

VT

andS

pV

.

(ii) Hence, using T dS=dU+pdV(conservation of energy with Uthe internal energy), showthat

lnp

lnV

T

lnp

lnV

S

=

(pV)

T

V

p1(U/V)T+ 1

(U/T)V

.

Hint:

lnplnV

T

= Vp

pV

T

13. By differentiatingIwith respect to , show that

I(, ) =

0

sin x

x exdx= tan1

+c().

Show that c() is constant (independent of) and hence, by considering the limits and 0, show that, if >0,

0

sin x

x dx=

2.

14. Let f(x) =x

0 et

2

dt2

and let g(x) =10[ex

2(t2+1)/(1 +t2)]dt.

Show thatf(x) +g(x) = 0.

Deduce that f(x) +g(x) =/4,

and hence that

0

et2

dt=

2 .

Comments and corrections may be sent by email to [email protected]

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