# 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
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