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ordinary differential equation (ODE) x2y′′(x) + xy′(x) − n2y(x) = 0, n = 0,1,2,3,…
(a) Given n > 0:
(i) Solve the ODE by trialling y(x) = xλ.
(ii) Use the Wronskian to prove that the solutions from part (i) are linearly independent.
(b) Given n = 0, use reduction of order to find the general solution to the ODE.
(a) Use the Weierstrass test to determine if the following series converges uniformly
.
(b) Consider (1 − x2)y′′(x) − xy′(x) + y(x) = 0.
(i) Solve the ODE using a power series expanded about x = 0
.
(ii) State the interval of convergence of any infinite series solutions you find.
(iii) Suppose y(0.2) = 1,y′(0.2) = 0. Without solving the initial value problem (IVP), what is the largest interval in x in which a unique solution is guaranteed for the IVP?
3 (15 following system of ODEs
x′(t) + 2x(t) − y(t) − 3 = 0, y′(t) + 2y(t) − x(t) + 1 = 0, x(0) = 0, y(0) = 0.
(a) Convert the system into a vector-valued IVP.
(b) Solve the system by matrix diagonalisation.
(a) Find the inverse Laplace transform of
(b) TheAssignment Project Exam Helpdisplacement of a damped mass-spring system, y(t), is governed by the ODE
y(0) = 0, y′(0) = 0.
(i) Give an interpretation of the forcing term and the initial conditions.
5 (13
function defined over [−π,π].
(a) Sketch f(x).
(b) Is f(x) even, odd or neither?
(c) Find g(x), the Fourier series representation of f(x) = f(x + 2π)?
(d) Given your answer to part (c):
(i) How do the Fourier coefficients in g(x) decay for large n?
(ii) Would you expect g(x) to converge uniformly? Briefly explain your answer.
(iii) For which x in [−π,π] does g(x) exhibit Gibbs phenomenon? What is the error of the truncated Fourier approximation at this x value?
(e) Consider the function Assignment Project Exam Help .
Without calculations, does the Fourier series of h(x) converge uniformly?
where
u(x,t) → 0 as x → ±∞
and
u(x,0) = δ2(x) + δ−2(x) and ∂tu(x,0) = 0.
(a) From first principles, find the Fourier transform of
(i) f(x) = δ2(x)
(ii) g(x) = δ−2(x)
(b) Use your results in part (a) to find the inverse Fourier transform of h(k) = cos2k.
(c) Find the IVP for the Fourier transform of u.
(d) Without calculations, briefly describe the remaining steps to find u.
7 (28
The temperature u(r,θ,t) of a metal semi-annulus plate is modelled by the heat equation
, 1 < r < 2, 0 ≤θ ≤π, t > 0.
(a) Consider the steady state temperature u(r,θ) subject to the boundary conditions
1 < r < 2,
1 < r < 2,
0 ≤θ ≤π,
, 0 ≤θ ≤π.
(i) Use separation of variables to show that the steady state heat equation reduces to:
,
where λ∈R is the separation constant.
, 1 < r < 2, 0 ≤θ ≤π.
u(r,0) = r, 1 ≤ r ≤ 2.
(i) Find the total heat of the annulus plate.
(ii) Without solving the governing PDE for u(r,t), sketch u(r,t) versus r, 1 ≤ r ≤ 2, 0 ≤ t < ∞ for the following four time states t:
0 < t1 < t2 < limt →∞.
(c) Suppose the formal series solution is a genuine solution to
, 0 ≤θ ≤π, t > 0.
Which of the series and its derivatives converge uniformly in the given domain?
MAST20030 Differential Equations Formulae Sheet
1) Laplace Transforms
1. f(t) ∞
(Lf)(s) = F(s) = f(t)e−st dt (Definition of Transform)
0
2. 1
3. tn
4. eat
5. sin(at)
6. cos(at)
7. Assignment Project Exam Helpsinh(at)
8. cosh(at)
11. f′′(t) s2F(s) − sf(0) − f′(0)
12. f(n)(t) snF(s) − n−1 sn−1−kf(k)(0)
k=0
(s-Shifting Theorem)
(a > 0, t-Shifting Theorem)
(Convolution)
2) Standard Limits
(i)(ii) lim rn = 0 (|r| < 1)
n→∞
(iii)(iv) lim n1/n = 1
n→∞
(vi) en log
p
3) The Generalised Harmonic Series (p-Series)
∞ 1 convergent if p > 1
is
divergent if p ≤ 1
4) Geometric Series
5) Fourier Series Formulae
6) Fourier Transforms Formulae
d dnxfn = (ik)nf(k), xnf(x) = indkn
7) Trigonometric and Hyperbolic Formulae
cos x + sin2 x = 1 cosh2 x − sinh2 x = 1
1 + tan2 x = sec2 x 1 − tanh2 x = sech2 x
cot2 x + 1 = cosec2 x coth2 x − 1 = cosech2 x
sin(x + y) = sinxcosy + cosxsiny sinh(x + y) = sinhxcoshy + coshxsinhy
cos(x + y) = cosxcosy − sinxsiny cosh(x + y) = coshxcoshy + sinhxsinhy
sin sinhxsinh cos coshxcosh sin sinhxcosh
sinAssignment Project Exam Help sinh
cos cosh
