18.034 Problem Set #2

Notation. � = d/dx.

1. Let

�

x/ x for x = 0, f(x) = | | �

k for x = 0,

where k is a constant. Show that no matter how the constant k is chosen, the differential equation

y� = f(x) has no solution on an interval containing the origin.

2. Suppose that f be a continuous bounded function for the entire real axis. If f� is continuous,

then show that the nonzero solution of the initial value problem of y� = yf(y) with y(0) = y0 =� 0

exists for all x. (You may need to assume the uniqueness theorem. )

3. Brikhoff-Rota, pp. 20, #9.

4. (The Ricatti equation) It is the differential equation of the form y� = a(x) + b(x)y + c(x)y2. In

general the Ricatti equation is not solvable by elementary means∗

. However,

(a) show that if y1(x) is a solution then the general solution is y = y1 + u, where u is the general

solution of a certain Bernoulli equation (cf. pset #1).

(b) Solve the Ricatti equation y� = 1 − x2 + y2 by the above method.

5. Let

Ly = y�� + y.

We are going to find the rest solution of the differential equation Ly = 3 sin 2x + 3 + 4ex. That is the

solution with u(0) = u�

(0) = 0.

(a) Find the general solution of Ly = 0.

(b) Solve Ly = 3 sin 2x, Ly = 3, and Ly = 4ex by use of appropriate trial solutions.

(c) Determine the constants in

y(x) = c1 cos x + c2 sin x − sin 2x + 3 + 2ex

to find the solution.

6. (Euler’s equi-dimensional equation) It is a differential equation of the form x2y��+pxy�

+qy = 0,

where p, q are constants.

(a) Show that the setting x = et changes the differential equation into an equation with constant

coefficients.

(b) Use this to find the general solution to x2y�� + xy� + y = 0.

(c) For which values of p, the general solutions of x2y�� + pxy� + 2y = 0 are defined for the entire

real axis (−∞, ∞)?

∗This was shown by Liouville in 1841.

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