1.5 Equations reducible to linear form
Theorem 1.5.1. For \(n \neq 0, 1\), the Bernoulli equation \(y' + p(t)y = q(t)y^n\) becomes linear under \(w = y^{1-n}\), on any interval where \(y > 0\): \[ w' + (1 - n)p(t)w = (1 - n)q(t) . \]
Proof. Differentiating and using the equation, \[ w' = (1 - n)y^{-n}y' = (1 - n)y^{-n}(qy^n - py) = (1 - n)(q - py^{1-n}) = (1 - n)(q - pw) . \] □
Theorem 1.5.2. Let \(q_2 \neq 0\) and consider the Riccati equation \(y' = q_0(t) + q_1(t)y + q_2(t)y^2\).
- (i)
- If \(y_1\) is one solution, then \(y = y_1 + 1/v\) is a solution if and only if \(v' = -(q_1 + 2q_2y_1)v - q_2\), a linear equation.
- (ii)
- The substitution \(y = -u'/(q_2u)\) turns the Riccati equation into the second-order linear equation \(u'' - \bigl (q_1 + q_2'/q_2\bigr )u' + q_0q_2u = 0\).
Proof. (i) Substitute \(y = y_1 + 1/v\): \(y_1' - v'/v^2 = q_0 + q_1y_1 + q_1/v + q_2(y_1^2 + 2y_1/v + 1/v^2)\). Since \(y_1\) is a solution, the terms without \(v\) cancel; multiplying the rest by \(-v^2\) gives \(v' = -(q_1 + 2q_2y_1)v - q_2\).
(ii) With \(y = -u'/(q_2u)\), \(y' = -\frac {u''}{q_2u} + \frac {u'^2}{q_2u^2} + \frac {q_2'u'}{q_2^2u}\) and \(q_2y^2 = \frac {u'^2}{q_2u^2}\), so the \(u'^2\) terms cancel in \(y' - q_2y^2\) and the equation becomes \(-\frac {u''}{q_2u} + \frac {q_2'u'}{q_2^2u} = q_0 - \frac {q_1u'}{q_2u}\). Multiplying by \(-q_2u\) gives the stated linear equation. □
Part (ii) shows that the non-linear Riccati equation and the general second-order linear equation are the same problem, which is why most Riccati equations, like most second-order linear equations with variable coefficients, have no elementary solution. Riccati equations reappear in optimal control and in the reduction of linear boundary value problems.
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