Maestra en FsicaFacultad de Fsica

Universidad Veracruzana

Spring 2015 CourseInstructor: J. Efran Rojas

MATHEMATICAL METHODS

Homework 1

Solve the first-order DEdu(x)dx

=u(x) 4xx u(x) .

Draw a direction field and some integral curves.

Find the Wronskian of two solutions of the given DE without solving the equationa)

x2d2u(x)dx2

+ xdu(x)dx

+(x2 2)u(x) = 0.

b)

(1 x2)d2u(x)dx2

2xdu(x)dx

+ (1 + )u(x) = 0.

Do you recognize these equations?

Show that if p(x) is differentiable and p(x) > 0 then the Wronskian of two solutions of ddx(p(x)dudx

)+

q(x)u(x) = 0 is W (x) = constp(x) .

Find a particular solution of the given DEs.a)

d2u(x)dx2

+ 2du(x)dx

+ u(x) = 3ex.

b)

4d2u(x)dx2

4du(x)dx

+ u(x) = 16ex/2.

Consider the DEx2

d2u(x)dx2

3xdu(x)dx

+ 4u(x) = x2 lnx, x > 0.

a) Obtain the functions u1 and u2 that satisfy the corresponding homogeneous DE.

b) Find a particular solution of the given inhomogeneous DE.

Delivery date: 02/11/2015

References

[1] W. E. Boyce and R. C. DiPrima, Elementary differential equations and boundary value problems, JohnWiley & Sons, Inc.,. NY (2001).

[2] Philippe Dennery and Andre Krzywicki, Mathematics for physicists, Dover Publications Inc., Mineola NY(1995).

February 5, 2015 1