Substitution Methods

Some equations \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=G(x,y)\) are neither directly integrable nor immediately separable. A suitable substitution \(u=F(x,y)\) can reveal a separable equation.

Differentiate the substitution correctly

Treat \(y\) as a function of \(x\). When \(F\) contains both variables, use the multivariable chain rule.

Substitution method

1. Set \(u=F(x,y)\) and compute \(\dfrac{\mathrm{d}u}{\mathrm{d}x}\).
2. Replace \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) using the original differential equation.
3. Rewrite every remaining occurrence of \(x\) and \(y\) in terms of \(u\) and the chosen independent variable.
4. Separate and integrate the resulting equation.
5. Substitute back for \(u\), then apply any initial condition.

If \(u=F(x)\) depends only on \(x\), the \(F_y\dfrac{\mathrm{d}y}{\mathrm{d}x}\) term is absent. If \(u=F(y)\), then \(\dfrac{\mathrm{d}u}{\mathrm{d}x}=F'(y)\dfrac{\mathrm{d}y}{\mathrm{d}x}\).