Internal problem ID [10527]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-3. Equations with
tangent.
Problem number: 29.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a y^{2}-b \tan \left (x \right ) y=c} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 187
dsolve(diff(y(x),x)=a*y(x)^2+b*tan(x)*y(x)+c,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\sec \left (x \right ) \left (-\left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) \left (b +\sqrt {4 a c +b^{2}}\right ) \sin \left (x \right )+\left (\sqrt {4 a c +b^{2}}-b +2\right ) \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right )\right )}{2 \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, -\frac {1}{2}+\frac {b}{2}, \sin \left (x \right )\right )\right ) a} \]
✓ Solution by Mathematica
Time used: 2.211 (sec). Leaf size: 608
DSolve[y'[x]==a*y[x]^2+b*Tan[x]*y[x]+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sin (x) \left (\left (-b^3+3 b^2+b-3\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+4 a c}+2\right ),\frac {1}{4} \left (-b+\sqrt {b^2+4 a c}+2\right ),\frac {3-b}{2},\cos ^2(x)\right )+\cos (x) \left ((b+1) \cos (x) (a c+b-1) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+4 a c}+6\right ),\frac {1}{4} \left (-b+\sqrt {b^2+4 a c}+6\right ),\frac {5-b}{2},\cos ^2(x)\right )+a i^{b+1} (b-3) c c_1 \cos ^b(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}+4\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}+4\right ),\frac {b+3}{2},\cos ^2(x)\right )\right )\right )}{a (b-3) (b+1) \left (\cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+4 a c}+2\right ),\frac {1}{4} \left (-b+\sqrt {b^2+4 a c}+2\right ),\frac {3-b}{2},\cos ^2(x)\right )-i i^b c_1 \cos ^b(x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}\right ),\frac {b+1}{2},\cos ^2(x)\right )\right )} \\ y(x)\to -\frac {c \sin (x) \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}+4\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}+4\right ),\frac {b+3}{2},\cos ^2(x)\right )}{(b+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}\right ),\frac {b+1}{2},\cos ^2(x)\right )} \\ y(x)\to -\frac {c \sin (x) \cos (x) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}+4\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}+4\right ),\frac {b+3}{2},\cos ^2(x)\right )}{(b+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{4} \left (b-\sqrt {b^2+4 a c}\right ),\frac {1}{4} \left (b+\sqrt {b^2+4 a c}\right ),\frac {b+1}{2},\cos ^2(x)\right )} \\ \end{align*}