2.6.5—Differential substitutions

Choose a new variable u=g(y) or u=g(x,y) so that the derivatives in the equation combine into du/dx or a simpler separable form. The substitution is useful only when it reduces the order or nonlinearity.

Differentiate the substitution explicitly, replace every occurrence consistently and recover y only after solving for u. Check any lost solutions introduced by division or square roots.

For an equation containing y′/y, set u=ln|y| so u′=y′/y; the transformed equation may become linear or directly integrable.

A substitution is not just a change of symbol: forgetting the chain-rule factor produces a different differential equation.