18.03 Practice Final Examination

Spring, 1999

Here is a copy of the information which will appear on the final exam itself.

Laplace Transform—Definition: L(f(t); s) = ∫_{0}^{∞} f(t)e^{−st}dt for Re s large.

Laplace Transform—Rules:

0. L(f(t); s) is unchanged if we alter any single value of f(t).

1. L is linear: L(f + g) = L(f) + L(g) and if c is constant then L(cf) = cL(f).

2. L essentially determines f: if f and g are piecewise continuous and L(f) = L(g), then f(a) = g(a) wherever both are continuous.

3. L preserves real and imaginary parts.

4. Scaling: For a > 0, L(f(t/a); s) = aL(f(t); as).

5. t-shift rule: L(f_{a}(t); s) = e^{−at}L(f(t); s), where f_{a}(t) = \{ \begin{matrix}f(t − a) & if t ≥ a \\ 0 & if t < a \end{matrix} .

6. s-shift rule: L(f(t); s − a) = L(e^{at}f(t); s).

7. t-derivative rule: L(f′(t); s) = sL(f(t); s) − f(0).

8. s-derivative rule: −\frac{d}{ds}L(f(t); s) = L(tf(t); s).

9. L(f ∗ g) = L(f)L(g).

10. L(δ) = 1.

Laplace Transform—Values:

L(1; s) = 1/s; L(e^{at}; s) = 1/(s − a)

L(t^{a−1}; s) = Γ(a)/s^{a}, a > 0; L(t^{n}; s) = n!/s^{n+1}, n = 0, 1, 2, …

Caution! The following formula has been corrected, and is different from an earlier

version of the practice final, when sin and cos were swapped.

L(sin(ωt); s) = \frac{ω}{s^{2} + ω^{2}} ; L(cos(ωt); s) = \frac{s}{s^{2} + ω^{2}}

Γ function: Γ(a) = ∫_{0}^{∞} e^{−t}t^{a−1}dt for a > 0.

aΓ(a) = Γ(a + 1), a > 0; Γ(n) = (n − 1)! for n a positive integer; Γ(1/2) = \sqrt{π}.

Heat equation:

\frac{∂u}{∂t} = k \frac{∂^{2}u}{∂x^{2}} with boundary conditions u(0) = u(π) = 0 has solutions u(x, t) = ∑_{n=1}^{∞} b_{n}e^{−kn^{2}t} sin(nx).

The first four problems all concern the initial value problem

\frac{dy}{dx} + xy = x, y(0) = 0,

but they are independent of one another.

1. This equation is separable. Find the general solution by separating variables, and then find the particular solution with y(0) = 0.

2. It is also a first order linear equation. Find an appropriate integrating factor and solve the same initial value problem again.

3. Use Euler’s method to estimate y(0.2) using a stepsize of 0.1.

4. Now draw an axis system (with range −3 ≤ x, y ≤ 3), sketch the isoclines for m = −1, 0, 1, and the direction field along them, and sketch a few solution curves (including the one through (0, 0)).

5. (a) Use the Laplace transform to solve the initial value problem ẍ − x = 0, x(0) = 0, ẋ(0) = 1.

(b) Express the solution to ẍ − x = r(t), x(0) = 0, ẋ(0) = 0, as a convolution using Duhamel’s principle. Write this convolution as an integral (from the definition).

6. Find the general solution of Ly = 0, where L is the differential operator L = (D+2I)^{5}.

7. Let A = [ \begin{matrix}a & 1 \\ −1 & 0 \end{matrix} ], where a is some real number, and consider the linear system ˙x⃗ = Ax⃗. Specify the values of a (and the answer may be “none”) for which the phase portrait exhibits each of the following. (Every number a should appear.)

(i) A nonconstant periodic orbit. (ii) A saddle. (iii) A stable spiral (not a center). (iv) An unstable spiral (not a center).

(v) A stable proper node. (vi) An unstable proper node. (vii) A stable improper node. (viii) An unstable improper node.

8. Consider the homogeneous linear equation y′′′ + y′′ + y′ + y = 0.

(a) Determine the normal modes (i.e., the exponential solutions) of this equation. (Hint: is −1 a root of the characteristic polynomial?)

(b) Write down three independent real solutions.

9. Use the Laplace transform to find the solution to the same homogeneous linear equation as in 8. (but written now using t for the independent variable and x for the independent variable), x^{(3)} + ẍ + ẋ + x = 0, for which x(0) = ẋ(0) = 0, ẍ(0) = 1.

10. Consider the linear autonomous system \{ \begin{matrix}ẋ & = & y \\ ẏ & = & 2x − y. \end{matrix} (a) Write down the second-order homogeneous linear equation which leads to this system by “anti-elimination.”

(b) Write the system as a vector-valued differential equation of the form ˙ x⃗ = Ax⃗. Find the eigenvalues of the matrix A, and for each eigenvalue find a nonzero eigenvector. Write down the normal modes (i.e., the exponential solutions).

(c) Write down a fundamental matrix for this system, and then normalize it to find e^{At}.

Not unrelated is:

10. Consider the nonlinear autonomous system \{ \begin{matrix}ẋ & = & y − 1 \\ ẏ & = & x^{2} − y. \end{matrix}

(a) Find the critical points of this system. Compute the Jacobian, evaluate it at each critical point, and decide the type of each critical point: saddle, unstable or stable spiral, unstable or stable node.

(b) Sketch the phase portrait of this system in a region including all the critical points. Mark the direction of travel on the trajectories.

11. Consider the Airy equation y′′ + xy = 0. Let ca(x) be the solution satisfying ca(0) = 1, ca′(0) = 0, and let sa(x) be the solution satisfying sa(0) = 0, sa′(0) = 1. The Wronskian of these two solutions is w(x) = ca(x)sa′(x) − ca′(x)sa(x).

Compute w′(x) and w(0), and then solve this initial value problem to find w(x). Explain why sa(a) ≠ 0 whenever ca(a) = 0. [Hint: suppose both were zero.]

(b) Explain why ca′(a) ≠ 0 whenever ca(a) = 0. [Hint: suppose both were zero.]

(c) Then use this information to show that if a, b are numbers such that a < b and ca(a) = ca(b) = 0 but ca(x) ≠ 0 for all x between a and b, then there is a number c such that a < c < b and sa(c) = 0. [This is analogous to the behavior of cos(x), sin(x): their zeros alternate.]

12. Let f(t) be the function which is periodic of period 2π and given by f(t) = t for −π/2 < t < π/2, f(t) = π − t for π/2 < t < 3π/2.

(a) Find the Fourier coefficients of f(t).

(b) Suppose a system controlled by the differential operator L = D^{2} + kI is driven by f(t). Find all solutions of Lx = f(t) which are periodic of period 2π. Your answer will depend upon k.

(c) If we have a bar of metal extending from x = 0 to x = π, whose heat distribution is

governed by the equation \frac{∂u}{∂t} = \frac{∂^{2}u}{∂x^{2}}, and we keep the ends fixed at 0 degrees, what is u(x, t)?