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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

Equilibrium Point Algebraic Equation Chain Rule Primitive Function Solution Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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