Internal problem ID [9686]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential
Functions
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right ) \left (y^{\prime }-y^{2}\right )+a \,\lambda ^{2} {\mathrm e}^{\lambda x}+b \,\mu ^{2} {\mathrm e}^{\mu x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 350
dsolve((a*exp(lambda*x)+b*exp(mu*x)+c)*(diff(y(x),x)-y(x)^2)+a*lambda^2*exp(lambda*x)+b*mu^2*exp(mu*x)=0,y(x), singsol=all)
\[ y \relax (x ) = -\frac {\left (\left (a b \lambda +\mu b a \right ) \left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )+a b \lambda c_{1}+a b \mu c_{1}\right ) {\mathrm e}^{\lambda x +\mu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_{1}+\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )}-\frac {\left (\left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right ) b^{2} \mu +c_{1} b^{2} \mu \right ) {\mathrm e}^{2 \mu x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_{1}+\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )}-\frac {\left (\left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right ) a^{2} \lambda +c_{1} a^{2} \lambda \right ) {\mathrm e}^{2 \lambda x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_{1}+\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )}-\frac {\left (\left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right ) b c \mu +c_{1} b c \mu \right ) {\mathrm e}^{\mu x}+1+\left (\left (\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right ) a c \lambda +c_{1} a c \lambda \right ) {\mathrm e}^{\lambda x}}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2} \left (c_{1}+\int \frac {1}{\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{\mu x}+c \right )^{2}}d x \right )} \]
✓ Solution by Mathematica
Time used: 29.224 (sec). Leaf size: 393
DSolve[(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*(y'[x]-y[x]^2)+a*\[Lambda]^2*Exp[\[Lambda]*x]+b*\[Mu]^2*Exp[\[Mu]*x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {-a e^{\lambda K[1]} \lambda ^2-b e^{\mu K[1]} \mu ^2+a e^{\lambda K[1]} y(x)^2+b e^{\mu K[1]} y(x)^2+c y(x)^2}{\left (e^{\lambda K[1]} a+b e^{\mu K[1]}+c\right ) \left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} y(x)+b e^{\mu K[1]} y(x)+c y(x)\right )^2}dK[1]+\int _1^{y(x)}\left (\frac {1}{\left (a e^{x \lambda } \lambda +b e^{x \mu } \mu +a e^{x \lambda } K[2]+b e^{x \mu } K[2]+c K[2]\right )^2}-\int _1^x\left (\frac {2 \left (-a e^{\lambda K[1]} \lambda ^2-b e^{\mu K[1]} \mu ^2+a e^{\lambda K[1]} K[2]^2+b e^{\mu K[1]} K[2]^2+c K[2]^2\right )}{\left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} K[2]+b e^{\mu K[1]} K[2]+c K[2]\right )^3}-\frac {2 a e^{\lambda K[1]} K[2]+2 b e^{\mu K[1]} K[2]+2 c K[2]}{\left (e^{\lambda K[1]} a+b e^{\mu K[1]}+c\right ) \left (a e^{\lambda K[1]} \lambda +b e^{\mu K[1]} \mu +a e^{\lambda K[1]} K[2]+b e^{\mu K[1]} K[2]+c K[2]\right )^2}\right )dK[1]\right )dK[2]=c_1,y(x)\right ] \]