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.
1
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