Math 334
Assignment 1
Due: 12 Noon on Thursday, September 21, 2006.
1. Direction Fields. Sketch the direction field for each of the following ordinary differential equations.
(a) dy
dx =x2−y2,(b) dy
dx =x2−y2
x2+y2,(c) dy
dx =x2
y2,(d) dy
dx =x−y.
2. Separable equations. Solve the following initial value problems.
(a) y′=x3(1 −y), y(0) = 3;
(b) dy/dθ =ysin θ, y (0) = √3;
(c) dy/dx =ex2/y2, y (0) = 1;
(d) dy/dx = (1 + y2)√1 + sin x, y(0) = 1.
3. Interval of definition. By looking at an initial value problem dy/dx =f(x, y) with y(x0) = y0,
it is not always possible to determine the domain of the solution y(x) or the interval over which the
function y(x) satisfies the differential equation.
(a) Solve the equation dy/dx =xy3.
(b) Give explicitly the solutions to the initial value problem with y(0) = 1; y(0) = 1/2; y(0) = 2.
(c) Determine the domains of the solutions in part (b).
(d) As found in part (c), the domains of the solutions depend on the initial conditions. For the initial
value problem dy/dx =xy3with y(0) = a, where a > 0, show that as a→0+, the domain
approaches the whole real line (−∞,∞), and as a→+∞, the domain shrinks to a single point.
4. Exact equations. For each of the following equations, find the most general function M(x, y) so that
the equation is exact:
(a) M(x, y)dx + (sec2y−x/y)dy = 0;
(b) M(x, y)dx + (sin xcos y−xy −e−y)dy = 0.
5. Integrating factors I. Solve the following differential equations:
(a) (3x2+y)dx + (x2y−x)dy = 0;
(b) (y2+ 2xy)dx −x2dy = 0.
6. Integrating factors I I. Find integrating factors of the form xnymto solve the following equations:
(a) (2y2−6xy)dx + (3xy −4x2)dy = 0;
(b) (12 + 5xy)dx + (6xy−1+ 3x2)dy = 0.
7. Integrating factors III. Show that if (∂N/∂x −∂M/∂y)/(xM −y N) depends only on the product
xy, that is
∂N/∂x −∂M/∂y
xM −yN =H(xy),
then the equation M(x, y )dx +N(x, y)dy = 0 has an integrating factor of the form µ=µ(xy). Give
the general formula for µ(xy).