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First Order Nonlinear Differential Equations

In a Nut Shell: There are two forms of first order, nonlinear d.e.’s that appear in

elementary differential equations -- those that are separable (easiest type) and those

that are nonseparable.

For separable forms, the key step is to place all the dependent variables and their derivatives

on one side of the d.e. and all the independent variables on the other side.

The general approach for nonseparable ones is to find a transformation that

yields a linear d.e. The type of transformation depends on the form of the nonlinear d.e.

Start with the easiest type -- first order, nonlinear, d.e.’s that are separable

dy/dx = y’ = f(x,y)

Suppose f(x,y) can be expressed as follows: f(x,y) = g(x) h(y)

Then the d.e. becomes dy/dx = g(x) h(y) and separation of variables gives

dy/ h(y) = g(x) dx which can be integrated directly to obtain y(x).

Nonseparable, first order nonlinear d.e.’s are more challenging.

The KEY STRATEGY: Use known transformations based on the form of the d.e.

Form of Differential Equation Transformation

dy/dx = f(ax + by + c)

Try v(x) = ax + by + c

Homogeneous equations

dy/dx = f(y/x)

Try v(x) = y/x

Bernoulli equations

y ’ + p(x) y = q(x) yⁿ

Try v(x) = y ^{1 - n}

So the first step is to identify the type of d.e. in order pick the relevant

substitution.

Click here for examples.

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