Riccati Equation

$y\text{'}\; =\; q\_0(x)\; +\; q\_1(x)\; \backslash ,\; y\; +\; q\_2(x)\; \backslash ,\; y^2$

$y\; =\; y\_1\; +\; u$

$y\_1\; +\; u$

$y\_1\text{'}\; +\; u\text{'}\; =\; q\_0\; +\; q\_1\; \backslash cdot\; (y\_1\; +\; u)\; +\; q\_2\; \backslash cdot\; (y\_1\; +\; u)^2,$

$y\_1\text{'}\; =\; q\_0\; +\; q\_1\; \backslash ,\; y\_1\; +\; q\_2\; \backslash ,\; y\_1^2$

$u\text{'}\; =\; q\_1\; \backslash ,\; u\; +\; 2\; \backslash ,\; q\_2\; \backslash ,\; y\_1\; \backslash ,\; u\; +\; q\_2\; \backslash ,\; u^2$

$u\text{'}\; -\; (q\_1\; +\; 2\; \backslash ,\; q\_2\; \backslash ,\; y\_1)\; \backslash ,\; u\; =\; q\_2\; \backslash ,\; u^2,$

$z\; =\; u^\{1-2\}\; =\; \backslash frac\{1\}\{u\}$

$y\; =\; y\_1\; +\; \backslash frac\{1\}\{z\}$

$z\text{'}\; +\; (q\_1\; +\; 2\; \backslash ,\; q\_2\; \backslash ,\; y\_1)\; \backslash ,\; z\; =\; -q\_2$

$y\; =\; y\_1\; +\; \backslash frac\{1\}\{z\}$

External link

• Riccati Equation at EqWorld: The World of Mathematical Equations.

Bibliography

• A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003.

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