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Separable Equations

If a first-order ordinary differential equation $F(y',y,x)=0$ can be rearranged so that only one variable, for instance $y$ , appears on the left-hand-side multiplying its derivative and the other, $x$ , appears only on the right-hand-side, then the equation is said to be `separated.''

$$\begin{split}& g(y)\ensuremath{\frac{d{y}}{d{x}}} = f(x)\ & g(y) dy = f(x) dx \end{split}$$ (19-6)

Each side of such an equation can be integrated with respect to the variable that appears on that side:

$$\int_{y(x_o)}^{y} g(\eta) d\eta = \int_{x_o}^x f(\xi) d \xi$$ (19-7)

if the initial value, $y(x_o)$ is known. If not, the equation can be solved with an integration constant $C_0$ ,

$$\int g(y) dy = \int f(x) dx + C_0$$ (19-8)

where $C_0$ is determined from initial conditions.

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ Example

(notebook Lecture-19)

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Using MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ 's Built-in Ordinary Differential Equation Solver

MATHEMATICA$^{\text{\scriptsize {\textregistered }}}$ has built-in exact and numerical differential equations solvers. DSolve takes a representation of a differential equation with initial and boundary conditions and returns a solution if it can find one. If insufficient initial or boundary conditions are specified, then ``integration constants'' are added to the solution.

While the accuracy of the first-order differencing scheme can be determined by comparison to an exact solution, the question remains of how to establish accuracy and convergence with the step-size $\delta$ for an arbitrary ODE. This is a question of primary importance and studied by Numerical Analysis.