Separable Scalar Initial Value Problems

  • Kenneth Eriksson
  • Donald Estep
  • Claes Johnson
Chapter

Abstract

We now consider the initial value problem for a scalar non-autonomous differential equation:
$$u'\left( x \right) = f\left( {u\left( x \right),x} \right)for0\angle x1,u\left( 0 \right) = {u_0},$$
(39.1)
in the special case when f (u(x), x) has the form
$$u'(x) = \frac{{h(x)}}{{g(u(x))}}for0\angle x1,u(0) = {u_0},$$
(39.2)
where h : ℝ → ℝ and g : ℝ → ℝ. We thus consider the initial value problem
$$u'(x) = \frac{{h(x)}}{{g(u(x))}}for0\angle x1,u(0) = {u_0},$$
(39.3)
where g : ℝ → ℝ and h : ℝ → ℝ are given functions, which we refer to as a separable problem, because the right hand side f (u(x), x) separates into the quotient of one function h(x) of x only, and one function g(u(x)) of u(x) only according to (39.2).

Keywords

Income 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Kenneth Eriksson
    • 1
  • Donald Estep
    • 2
  • Claes Johnson
    • 1
  1. 1.Department of MathematicsChalmers University of TechnologyGöteborgSweden
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA

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