Internal problem ID [10435]
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: 17.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-a \,{\mathrm e}^{\left (2 \lambda +\mu \right ) x} y^{2}-\left (b \,{\mathrm e}^{\left (\lambda +\mu \right ) x}-\lambda \right ) y=c \,{\mathrm e}^{\mu x}} \]
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 87
dsolve(diff(y(x),x)=a*exp((2*lambda+mu)*x)*y(x)^2+(b*exp((lambda+mu)*x)-lambda)*y(x)+c*exp(mu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\mu x} \left (\sqrt {4 b^{2} a c -b^{4}}\, \tan \left (\frac {\sqrt {4 b^{2} a c -b^{4}}\, \left ({\mathrm e}^{x \left (\lambda +\mu \right )} b +c_{1} \lambda +c_{1} \mu \right )}{2 b^{2} \left (\lambda +\mu \right )}\right )-b^{2}\right ) {\mathrm e}^{-x \left (\lambda +\mu \right )}}{2 a b} \]
✓ Solution by Mathematica
Time used: 6.375 (sec). Leaf size: 349
DSolve[y'[x]==a*Exp[(2*\[Lambda]+\[Mu])*x]*y[x]^2+(b*Exp[(\[Lambda]+\[Mu])*x]-\[Lambda])*y[x]+c*Exp[\[Mu]*x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{\lambda (-x)} \left (b^2 e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )-b (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )-4 a c e^{x (\lambda +\mu )} \left (\pi +i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}-1\right )\right )\right )}{2 a (\lambda +\mu ) \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \left (\pi -i c_1 \left (e^{\sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}}+1\right )\right )} y(x)\to \frac {e^{\lambda (-x)} \left (-(\lambda +\mu ) e^{-x (\lambda +\mu )} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}} \tanh \left (\frac {1}{2} \sqrt {\frac {\left (b^2-4 a c\right ) e^{2 x (\lambda +\mu )}}{(\lambda +\mu )^2}}\right )-b\right )}{2 a} \end{align*}