Internal problem ID [9780]
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]
\[ \boxed {y^{\prime }-y^{2} a -b \tan \left (x \right ) y-c=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 359
dsolve(diff(y(x),x)=a*y(x)^2+b*tan(x)*y(x)+c,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (\left (-c_{1} b +c_{1} \right ) \operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )+\left (1-b \right ) \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )\right ) \sin \left (x \right )^{3}+\left (\left (\left (\sqrt {4 a c +b^{2}}\, c_{1} +c_{1} \right ) \operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )+\left (1+\sqrt {4 a c +b^{2}}\right ) \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )\right ) \cos \left (x \right )^{2}+\left (c_{1} b -c_{1} \right ) \operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )+\left (b -1\right ) \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )\right ) \sin \left (x \right )+\left (\left (-\sqrt {4 a c +b^{2}}\, c_{1} +c_{1} b -2 c_{1} \right ) \operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )+\left (-\sqrt {4 a c +b^{2}}+b -2\right ) \operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}+\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )\right ) \cos \left (x \right )^{2}}{2 \cos \left (x \right ) \left (\sin \left (x \right )^{2}-1\right ) a \left (\operatorname {LegendreQ}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right ) c_{1} +\operatorname {LegendreP}\left (\frac {\sqrt {4 a c +b^{2}}}{2}-\frac {1}{2}, \frac {b}{2}-\frac {1}{2}, \sin \left (x \right )\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.153 (sec). Leaf size: 599
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 (-(b-3) (b-1) (b+1) \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 (2 x) \, _2\tilde {F}_1\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 )}{4 \, _2\tilde {F}_1\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 (2 x) \, _2\tilde {F}_1\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 )}{4 \, _2\tilde {F}_1\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*}