where A, B, C are functions of x, is, by the substitution
| η = − | 1 | dy |
| Cy | dx |
reduced to the linear equation
| d²y | − (B + | 1 | dC | ) | dy | + ACy = 0. |
| dx² | C | dx | dx |
The equation
| dη | = A + Bη + Cη², |
| dx |
known as Riccati’s equation, is transformed into an equation of the same form by a substitution of the form η = (aY + b)/(cY + d), where a, b, c, d are any functions of x, and this fact may be utilized to obtain a solution when A, B, C have special forms; in particular if any particular solution of the equation be known, say η0, the substitution η = η0 − 1/Y enables us at once to obtain the general solution; for instance, when
| 2B = | d | log ( | A | ), |
| dx | C |
a particular solution is η0 = √(-A/C). This is a case of the remark, often useful in practice, that the linear equation
| φ(x) | d²y | + ½ | dφ | dy | + μy = 0, |
| dx² | dx | dx |