Internal problem ID [10545]
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]
\[ \boxed {\left (a \tan \left (x \lambda \right )+b \right ) y^{\prime }-y^{2}-k \tan \left (\mu x \right ) y=-d^{2}+k d \tan \left (\mu x \right )} \]
✓ Solution by Maple
Time used: 0.015 (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)
\[ y \left (x \right ) = -d -\frac {{\mathrm e}^{\int \frac {\tan \left (\mu x \right ) k}{a \tan \left (\lambda x \right )+b}d x} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (1+\tan \left (\lambda x \right )^{2}\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \arctan \left (\tan \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{\int \frac {{\mathrm e}^{\int \frac {\tan \left (\mu x \right ) k}{a \tan \left (\lambda x \right )+b}d x} \left (a \tan \left (\lambda x \right )+b \right )^{-\frac {2 a d}{\lambda \left (a^{2}+b^{2}\right )}} \left (1+\tan \left (\lambda x \right )^{2}\right )^{\frac {a d}{\lambda \left (a^{2}+b^{2}\right )}} {\mathrm e}^{-\frac {2 d b \arctan \left (\tan \left (\lambda x \right )\right )}{\lambda \left (a^{2}+b^{2}\right )}}}{a \tan \left (\lambda x \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 130.719 (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]
\[ \text {Solve}\left [\int _1^x\frac {e^{-\int _1^{K[2]}\frac {\sec (\mu K[1]) (2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+k \sin (\lambda K[1]-\mu K[1])-k \sin (\lambda K[1]+\mu K[1]))}{2 (b \cos (\lambda K[1])+a \sin (\lambda K[1]))}dK[1]} (d \cos (\lambda K[2]-\mu K[2])-y(x) \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])+k \sin (\lambda K[2]-\mu K[2])-k \sin (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) y(x))}{k \mu (b \cos (\lambda K[2]-\mu K[2])+b \cos (\lambda K[2]+\mu K[2])+a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {e^{-\int _1^x\frac {\sec (\mu K[1]) (2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+k \sin (\lambda K[1]-\mu K[1])-k \sin (\lambda K[1]+\mu K[1]))}{2 (b \cos (\lambda K[1])+a \sin (\lambda K[1]))}dK[1]}}{k \mu (d+K[3])^2}-\int _1^x\left (\frac {e^{-\int _1^{K[2]}\frac {\sec (\mu K[1]) (2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+k \sin (\lambda K[1]-\mu K[1])-k \sin (\lambda K[1]+\mu K[1]))}{2 (b \cos (\lambda K[1])+a \sin (\lambda K[1]))}dK[1]} (-\cos (\lambda K[2]-\mu K[2])-\cos (\lambda K[2]+\mu K[2]))}{k \mu (d+K[3]) (b \cos (\lambda K[2]-\mu K[2])+b \cos (\lambda K[2]+\mu K[2])+a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}-\frac {e^{-\int _1^{K[2]}\frac {\sec (\mu K[1]) (2 d \cos (\lambda K[1]-\mu K[1])+2 d \cos (\lambda K[1]+\mu K[1])+k \sin (\lambda K[1]-\mu K[1])-k \sin (\lambda K[1]+\mu K[1]))}{2 (b \cos (\lambda K[1])+a \sin (\lambda K[1]))}dK[1]} (d \cos (\lambda K[2]-\mu K[2])-K[3] \cos (\lambda K[2]-\mu K[2])+d \cos (\lambda K[2]+\mu K[2])-\cos (\lambda K[2]+\mu K[2]) K[3]+k \sin (\lambda K[2]-\mu K[2])-k \sin (\lambda K[2]+\mu K[2]))}{k \mu (d+K[3])^2 (b \cos (\lambda K[2]-\mu K[2])+b \cos (\lambda K[2]+\mu K[2])+a \sin (\lambda K[2]-\mu K[2])+a \sin (\lambda K[2]+\mu K[2]))}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]