There are a number of equations known as the Riccati differential equation. The most common is
![z^2w^('')+[z^2-n(n+1)]w=0](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_tjHlUJwXnkX1f1rP8uyEhYWppAg_LE55L2dWMnLeEjhl_n52c0rB1wiCfrSku_MoFMycis3ePgVV4Bky4HKpSbseFkJMdeRBrKaAbql_VFSifWotesgD7enqpCk6I7hq0ZjQFlNPiPFPf-YMI8c_HDfvC-5q8LMarL_ax-_cxkleE=s0-d) |
(1)
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(Abramowitz and Stegun 1972, p. 445; Zwillinger 1997, p. 126), which has solutions
 |
(2)
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and 
are
Another Riccati differential equation is
 |
(3)
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which is solvable by algebraic, exponential, and logarithmic functions only when 
, for 
, 1, 2, ....
, for , 1, 2, ....
Yet another Riccati differential equation is
 |
(4)
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where 
(Boyce and DiPrima 1986, p. 87). The transformation
(Boyce and DiPrima 1986, p. 87). The transformation |
(5)
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leads to the second-order linear homogeneous equation
![R(z)y^('')-[R^'(z)+Q(z)R(z)]y^'+[R(z)]^2P(z)y=0.](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uFT9_7OD9t7vObfSX6VChSsEh7ZwocZKOQ7W5yBjyL5XQTFZbcVKAHm_fDldtrUjZU4mFq6g_IpXCAehE1gWJJXNsERwIsfafwlTwcY1V1hhVL9At6lpPRn_CWzTQts-LR3KMsMx_J9HqaNt3fm-z9wP6WCQncZdDI1VHwefcDrHQ=s0-d) |
(6)
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If a particular solution 
to (4) is known, then a more general solution containing a single arbitrary constant can be obtained from
to (4) is known, then a more general solution containing a single arbitrary constant can be obtained from |
(7)
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where 
is a solution to the first-order linear equation
is a solution to the first-order linear equation![v^'=-[Q(z)+2R(z)w_1(z)]v-R(z)](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vYdCyegE3dOWGrF3RXjNKL5nmCPoFbcNofJaYotmLPdGoOA8FmYq5tfD7NuEc358AoBVKNYpuD8JyLH5OAiJU0AmijzmV2GTs75rQClVMWbB8Fs56KWczb6QGrDA4Ya0etfCB8xDmjUOfF8X2ddUJKCafJk-uPWNnAw6svme9Rh_hH=s0-d) |
(8)
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(Boyce and DiPrima 1986, p. 87). This result is due to Euler in 1760.
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