Internal problem ID [10523]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.6-2. Equations with
cosine.
Problem number: 25.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left (\cos \left (\lambda x \right ) a +b \right ) y^{\prime }-y^{2}-c \cos \left (x \mu \right ) y=-d^{2}+c d \cos \left (x \mu \right )} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 268
dsolve((a*cos(lambda*x)+b)*diff(y(x),x)=y(x)^2+c*cos(mu*x)*y(x)-d^2+c*d*cos(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-d \left (\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (x \mu \right )}{a \cos \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cos \left (x \lambda \right )+b}d x \right )+d c_{1} -{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (x \mu \right )}{a \cos \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{\int \frac {{\mathrm e}^{\frac {c \left (\int \frac {\cos \left (x \mu \right )}{a \cos \left (x \lambda \right )+b}d x \right ) \sqrt {a^{2}-b^{2}}\, \lambda -4 d \,\operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {x \lambda }{2}\right )}{\sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, \lambda }}}{a \cos \left (x \lambda \right )+b}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 12.31 (sec). Leaf size: 289
DSolve[(a*Cos[\[Lambda]*x]+b)*y'[x]==y[x]^2+c*Cos[\[Mu]*x]*y[x]-d^2+c*d*Cos[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right ) (-d+c \cos (\mu K[2])+y(x))}{c \mu (b+a \cos (\lambda K[2])) (d+y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right )}{c \mu (b+a \cos (\lambda K[2])) (d+K[3])}-\frac {\exp \left (-\int _1^{K[2]}\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right ) (-d+c \cos (\mu K[2])+K[3])}{c \mu (b+a \cos (\lambda K[2])) (d+K[3])^2}\right )dK[2]-\frac {\exp \left (-\int _1^x\frac {2 d-c \cos (\mu K[1])}{b+a \cos (\lambda K[1])}dK[1]\right )}{c \mu (d+K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]