First Order Nonlinear D.E.

Example: Solve the following d.e. dy/dx = ( 9x + y + 10)²

Note: The form of D.E. is dy/dx = f( ax + by + c )

Strategy: Try the substitution: v = 9x + y + 10 then dv/dx = 9 + dy/dx

so that dv/dx - 9 = v² or dv/dx = 9 + v² which is separable

In separable form dv / (9 + v² ) = dx which can be integrated to obtain v.

Hint: Use v = 3 tan θ to aid in the integration. The result is

tan^-1 (v/3) = 3 x + C or v/3 = tan ( 3 x + C )

But v = 9x + y + 10 so y = 3 tan (3 x + C) ˗ 9x ˗ 10 (result)

Example: Solve the following d.e. x² dy/dx = xy + x² e^y/x

This nonlinear d.e. is not in standard form. So first divide each term by x² to obtain

dy/dx = y/x + e^y/x

Note: The Form of D.E. is dy/dx = f( y/x ) “Homogeneous form”

Strategy: Try the transformation v = y/x or y = v x and dy/dx = x dv/dx + v

The result is x dv/dx + v = v + e^v or x dv/dx = e^v which is separable

e^-v dv = dx/x which after integration gives ˗ e^-v = ln x + C₁ or

e ^{- v} = C₂ - ln ( 1/x )

Now v = y/x so e ^–y/x = C₂ - ln ( 1/x ) next take the log of both sides

( - y/x) = ln [C₂ - ln x ] Thus

y(x) = x ln [ 1 / (C₂ - ln x ) ] (result)

Click here for an example of a d.e. of the Bernoulli form.

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