Mathematics and Statistics · Ch 8 — Differential Equation and Applications

Solution by Variables Separable

A solution of a differential equation is a relation between $x$ and $y$ (free of derivatives) that satisfies the equation. A general solution carries as many arbitrary constants as the order of the equation; fixing those constants using given conditions (initial/boundary values) gives a particular solution.

The simplest solvable first-order equations are those in which the variables can be separated — written so that everything involving $y$ (together with $dy$ ) is on one side and everything involving $x$ (together with $dx$ ) on the other. An equation of the form

\frac{dy}{dx} = f(x)\,g(y)

can be rearranged to

\frac{dy}{g(y)} = f(x)\,dx,

and then integrated on both sides to obtain the general solution.

Worked illustration. Solve $\dfrac{dy}{dx} = \dfrac{x}{y}$ .

Separate: $y\,dy = x\,dx$ . Integrate both sides:

\int y\,dy = \int x\,dx \;\Longrightarrow\; \frac{y^2}{2} = \frac{x^2}{2} + c_1 \;\Longrightarrow\; y^2 - x^2 = c,

where $c = 2c_1$ is the single arbitrary constant.

Note

One Constant Is Enough …

Definition 6General and particular solution

A general solution contains arbitrary constants equal in number to the order of the equation; substituting given conditions to fix those constants yi …

Definition 7Variables separable form

A first-order equation $\frac{dy}{dx}=f(x)g(y)$ that can be rewritten as $\frac{dy}{g(y)}=f(x)\,dx$ and solved by integratin …

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