Internal problem ID [9792]
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: 37.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left (a \tan \left (\lambda x \right )+b \right ) y^{\prime }-y^{2}-k \tan \left (\mu x \right ) y+d^{2}-k d \tan \left (\mu x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 213
dsolve((a*tan(lambda*x)+b)*diff(y(x),x)=y(x)^2+k*tan(mu*x)*y(x)-d^2+k*d*tan(mu*x),y(x), singsol=all)
\[ y \relax (x ) = -d -\frac {{\mathrm e}^{\int \frac {k \tan \left (\mu x \right )}{a \tan \left (\lambda x \right )+b}d x} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (1+\tan ^{2}\left (\lambda x \right )\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \arctan \left (\tan \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{\int \frac {{\mathrm e}^{\int \frac {k \tan \left (\mu x \right )}{a \tan \left (\lambda x \right )+b}d x} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (1+\tan ^{2}\left (\lambda x \right )\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \arctan \left (\tan \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{a \tan \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 57.824 (sec). Leaf size: 800
DSolve[(a*Tan[\[Lambda]*x]+b)*y'[x]==y[x]^2+k*Tan[\[Mu]*x]*y[x]-d^2+k*d*Tan[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[6]}\frac {\sec (\mu K[5]) (2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+k \sin (\lambda K[5]-\mu K[5])-k \sin (\lambda K[5]+\mu K[5]))}{2 (b \cos (\lambda K[5])+a \sin (\lambda K[5]))}dK[5]} (d \cos (\lambda K[6]-\mu K[6])-y(x) \cos (\lambda K[6]-\mu K[6])+d \cos (\lambda K[6]+\mu K[6])+k \sin (\lambda K[6]-\mu K[6])-k \sin (\lambda K[6]+\mu K[6])-\cos (\lambda K[6]+\mu K[6]) y(x))}{k \mu (b \cos (\lambda K[6]-\mu K[6])+b \cos (\lambda K[6]+\mu K[6])+a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6])) (d+y(x))}dK[6]+\int _1^{y(x)}\left (\frac {e^{-\int _1^x\frac {\sec (\mu K[5]) (2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+k \sin (\lambda K[5]-\mu K[5])-k \sin (\lambda K[5]+\mu K[5]))}{2 (b \cos (\lambda K[5])+a \sin (\lambda K[5]))}dK[5]}}{k \mu (d+K[7])^2}-\int _1^x\left (\frac {e^{-\int _1^{K[6]}\frac {\sec (\mu K[5]) (2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+k \sin (\lambda K[5]-\mu K[5])-k \sin (\lambda K[5]+\mu K[5]))}{2 (b \cos (\lambda K[5])+a \sin (\lambda K[5]))}dK[5]} (-\cos (\lambda K[6]-\mu K[6])-\cos (\lambda K[6]+\mu K[6]))}{k \mu (d+K[7]) (b \cos (\lambda K[6]-\mu K[6])+b \cos (\lambda K[6]+\mu K[6])+a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6]))}-\frac {e^{-\int _1^{K[6]}\frac {\sec (\mu K[5]) (2 d \cos (\lambda K[5]-\mu K[5])+2 d \cos (\lambda K[5]+\mu K[5])+k \sin (\lambda K[5]-\mu K[5])-k \sin (\lambda K[5]+\mu K[5]))}{2 (b \cos (\lambda K[5])+a \sin (\lambda K[5]))}dK[5]} (d \cos (\lambda K[6]-\mu K[6])-K[7] \cos (\lambda K[6]-\mu K[6])+d \cos (\lambda K[6]+\mu K[6])-\cos (\lambda K[6]+\mu K[6]) K[7]+k \sin (\lambda K[6]-\mu K[6])-k \sin (\lambda K[6]+\mu K[6]))}{k \mu (d+K[7])^2 (b \cos (\lambda K[6]-\mu K[6])+b \cos (\lambda K[6]+\mu K[6])+a \sin (\lambda K[6]-\mu K[6])+a \sin (\lambda K[6]+\mu K[6]))}\right )dK[6]\right )dK[7]=c_1,y(x)\right ] \]