Differential equations, comprehensive exam topics and sample questions Topics covered ODE s: Chapters -5, 7, from Elementary Differential Equations by Edwards and Penney, 6th edition.. Exact solutions of first and second order ODEs: separable, linear, exact, integrating factor, of form y = F (y, y ) and y = F (x, y ), linear n-th order, variation of parameters. 2. Series solutions: various singularities, method of Frobenius. 3. Laplace transform methods: Table will be given (see attached). Transforms for periodic, functions, step function, and delta functions. 4. Linear systems: exact solutions; fundamental matrix; inhomogeneous systems; stability; phase portraits, classification of equilibria. 5. Stability and phase analysis: equilibria, stability, phase portrait, Hopf bifurcation. PDE s: Chapters 8,9 from Edwards&Penney; Chapters,2,4,5,6, 7, 4. from Partial Differential Equations by Walter Strauss. Derivation of diffusion/heat equation and wave equation 2. Maximum principle 3. Heat and wave equation on entire line; heat kernel 4. Separation of Variables, Strurm-Liouville problems, Eigenfunction Expansions (Chap. 8,9 of E-P or chap 4-5 of Strauss) 5. Method of Characteristics, including nonlinear PDE s and shock formation (chap. 4. from Strauss) 6. Fourier transform, Laplace transform. Vector calculus: see Adams, multivariable calculus.. Divergence, Stokes, Gauss theorems; computing fluxes. 2 Some sample exam questions. Find the general solution to each one the following ODE s: (a) y 6y + 9y = exp( 3x) (b) y + y = tanh(x) (c) y y y y = t (d) y = y2 x 2xy (e) y = yy
2. Let M =. (a) Compute exp (tm). (b) Determine the solution to ẏ = My subject to y() = (, 2, ). (c) Determine the general solution to y = My + e t ; y() = (, 2, ). (d) Solve the following initial value problem using the Laplace transform, y + 4y = δ (t π) + δ(t 2π) + δ (t 3π), y () =, y () =. Here δ is the Dirac delta function. Sketch the resulting solution. (e) Use Laplace transform to find the solution to tx (t) + 2x (t) + tx = ; x() =, x () =. (f) Use Laplace transform and the convolution to show that the solution to x + 2x + x = f(t); x() = = x () is given by 3. Consider the Hermite ODE, x(t) = t τe τ f(t τ)dτ. u 2xu + λu =, < x <. () (a) Compute the recursion relation for the power series solution u = a nx n. (b) Show that () has a polynomial solution of degree n whenever λ = 2n. Denote such a solution by H n (x). (c) Show that there exists a function w (x) > such that H n (x) H m (x) w(x) = whenever n m. Find w (x). 4. Consider the ODE system modelling a certain two-species interaction, x = x ( (x + y)) y, x, y. = y (2 (x + y)) Here, x and y represent two competing species. a) Plot the nullclines of the system and find the equilibria points. b) Classify the stability of each of the equilibria and sketch the phase portrait of the system. c) What is the eventual state of the system, in the limit t, if initially x > and y >? 5. Consider the system ẋ = y + µx + xy 2, ẏ = x + µy x 2. a. Analyse the stability of the zero equilibrium and classify any bifurcations that occur as µ is varied. b. Find all equilibria when µ = and classify their stability. c. With µ =, sketch a phase plot diagram, labelling nullclines and equilibrium points. 2
6. (a) Let u(x, t) be the vertical displacement of an elastic homogenous string undergoing small transverse vibrations. Carefully derive the wave equation ρu tt = T u xx, where T is the tension magnitude and ρ is its linear density. (b) In the derivation above, you ve neglected the gravity. How does the equation change if you incorporate the gravity acting in the vertical direction? 7. (a) Carefully derive the one-dimensional diffusion equation. (b) Suppose that a pollutant is diffusing in a river. The river moves at speed v from left to right. Derive the PDE describing this situation. 8. Solve the Wave equation u tt = u xx on the entire plane subject to the initial conditions u (x, ) = φ(x), u t (x, ) = where φ(x) is given by, if x < φ(x) =, if x. 9.. Sketch your solution for t =,.5, and.5. (a) State the maximum principle for the Heat equation. (b) Consider the heat equation with drift: u t = u xx + vu x where v is some constant. Does the maximum principle also hold for this PDE? Explain why or why not. (c) Does the maximum principle still hold for the heat equation u t = u xx if = is replaced by? by? (a) What is the general solution to the Wave equation u tt = u xx? What is its solution when the u(x, ) = φ(x) and u t (x, ) = ψ(x)? (b) Consider the PDE 3u tt +u xx 4u xt =. Is it elliptic, parabolic or hyperbolic? Find its general solution. HINT: 3t 2 + x 2 4xt = (x 2t) 2 t 2.. Show that the solution to the problem u t = u xx + f(x, t), u(x, ) = φ(x), u(, t) = h(t), u(, t) = g(t) is unique. 2. A rod is is heated from the right end and cooled at the left end. Initially, the temperature is zero. This situation can be modelled as: (a) Determine u(x, t). u t = u xx ; u(, t) =, u(, t) = u(x, ) =. (b) Let v(x) = u(x, ) = lim t u(x, t). Determine v(x). (c) Show that u(x, t) v(x) for all t >, x (, ) 3
3. Let D = (x, y) : y > and x 2 + y 2 < }, be an upper half-disk and suppose that u = inside D and moreover u has the following boundary conditions: u = on the right-bottom boundary y = and x > ; n u = on the left-bottom boundary y = and x < ; u = on the top boundary (x, y) = (cos θ, sin θ). Find the solution for u in terms of an appropriate infinite series. 4. Consider the system u t + (2 u)u x = ; u(x, ) = + tanh(x). (a) Determine the characteristic curves for this ODE. (b) Show that the solution develops a shock and compute the time t = t s at which the shock first occurs. (c) Sketch the solution profile u(x, t) for t =,.5,,.5. 5. Consider the PDE u t uu x = subject to the initial condition u(x, ) = φ(x) where φ(x) is a tent function given by x, x < φ(x) =, x. (a) Solve for the characteristics and sketch them. (b) Sketch the solution for t =,.5,,.5. (c) Does the solution exhibit a shock? If yes, determine t shock, the time when the shock first occurs. 6. Solve the PDE u t + 2xu x = t subject to the condition u(x, ) = x 2. 7. The temperature u(x, t) in a rod of length L satisfies u t = u xx with boundary conditions u x (, t) =, u (L, t) = and initial conditions u (x, ) = x. a) What is the steady state solution? b) Determine an expansion for u (x, t) in terms of appropriate eigenfunctions. Evaluate all coefficients explicitly. 8. Let λ < λ 2 < be the eigenvalues and y, y 2,... the corresponding eigenfunctions of the following boundary value problem on [, ]: y + λy =, y () = = 2y () + y () a) Derive an implicit formula for λ. Give an approximate formula for λ n for large n. b) Obtain a formula for the eigenfunctions y n. What orthogonality relation do they satisfy? c) Consider the boundary value problem u + µu = f(x) with u () = = 2u () + u (), µ >. Suppose that f can be expanded in terms of y n (x) as f = c n y n (x) for some coefficients c n. Derive the solution to this problem in terms of the eigenfunctions y n. 9. Solve the following problem for the Laplace s equation on the exterior of the unit disk: u rr + r u r + r u 2 θθ =, < θ 2π, r > u r (, θ) = 2 cos θ 3 sin 3θ u (r, θ) is bounded for r. 4
2. The telegraph equation is given by c 2 2 u x 2 = 2 u t 2 + 2b u t + au. u(x, ) = f(x), u (x, ) = g(x), t < x <, t >. (a) Set U(ω, t) = 2π u(x, t)eiωx dx and solve for U. (b) In the special case where b 2 = a solve for u(x, t). You may find the following Fourier transforms useful for this problem Where, 2π eiωa = F(δ(x a)), sin(aω) = F(v(x)), π ω cos(ωa) = F((δ(x + a) + δ(x a))) π v(x) = F(f(x)) = 2π L(f(t)) = if x a if x a f(x)e iωx dx f(t)e st dt 2. Let Ω be the top half of unit disk, Ω = (r cos θ, r sin θ) : < r <, θ (, π)}. Consider the following Green s function: 22. G = δ( x x ) inside Ω n G = on bottom boundary of Ω (y = ) G = on top (circle) boundary of Ω (r = ) (a) Write down an explicit formula for G. (hint: try the method of images) (b) Solve the following problem in terms of the Green s function you just found: u = inside Ω n u = f(x) on bottom boundary of Ω (y = ) u = g(θ) on top (circle) boundary of Ω (r = ) Simplify your final answer as much as you can. (a) Suppose that u = for all x R 2. Show that for any smooth domain D, and for any x D, we have: u(x ) = n ln x x u(x) ln x x n u(x)} ds(x). 2π D 5
(b) Deduce from part (a) that u(, ) = 2π 2π u(cos(θ), sin (θ))dθ. 23. Consider the PDE u t = u xx subject to initial condition u(x, ) = cos(x). By solving this PDE in two distinct ways, find the value of e ay2 cos(y)dy with a >. 24. (a) Briefly state Stokes, Green and Divergence theorems. (b) A smooth surface S lies above the plane z = and has as its boundary the circle x 2 + y 2 = 4y in the plane z =. This circle is also the boundary of a disk D in that plane. The volume of the 3-dimensional region R bounded by S and D is cubic units. Find the flux of F (x, y, z) = (x + x 2 y)i + (y xy 2 )j + (z + 2x + 3y)k through S in the direction outward from R. (c) Let F = 2xzi + e z2 j+ ( x 2 x cos (y) ) k. Using one of the three theorems stated in Part a, find F dr where c is the boundary of the unit sphere intersected with the xz plane, traversed c in the couterclockwise direction when viewed from (,, ). (d) Find the flux of F = xy 2 i + x 2 yj + k outward through the hemispherical surface x 2 + y 2 + z 2 = 4, z. Hint: You can use the Divergence theorem to simplify the computation. 6
Table of Laplace Transforms (will be given during the exam) f(t) = L F (s)} F (s) = Lf(t)} e at s a, t n n positive integer t p, p > sin(at) cos(at) t n e at, n positive integer H(t) s, s > s > a n! s n+, s > Γ(p + ) s p+, s > a s 2 + a 2, s > s s 2 + a 2, s > n! (s a) n+, s > a e cs s, s > H(t)f(t c) e cs F (s) e ct f(t) F (s c) t f(ct) f(t τ)g(τ) dτ F ( s ) c F, c > c (s)g(s) δ(t c) e cs f (n) (t) s n F (s) s n f() f (n ) () H(t) is the Heaviside function 7