Internal problem ID [10428]
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: 20.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {\left ({\mathrm e}^{\lambda x} a +b \,{\mathrm e}^{x \mu }+c \right ) y^{\prime }-y^{2}-k \,{\mathrm e}^{\nu x} y=-m^{2}+k m \,{\mathrm e}^{\nu x}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 202
dsolve((a*exp(lambda*x)+b*exp(mu*x)+c)*diff(y(x),x)=y(x)^2+k*exp(nu*x)*y(x)-m^2+k*m*exp(nu*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {-m \left (\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )+c_{1} m -{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{\int \frac {{\mathrm e}^{k \left (\int \frac {{\mathrm e}^{\nu x}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )-2 m \left (\int \frac {1}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x \right )}}{{\mathrm e}^{x \lambda } a +b \,{\mathrm e}^{x \mu }+c}d x -c_{1}} \]
✓ Solution by Mathematica
Time used: 16.545 (sec). Leaf size: 358
DSolve[(a*Exp[\[Lambda]*x]+b*Exp[\[Mu]*x]+c)*y'[x]==y[x]^2+k*Exp[\[Nu]*x]*y[x]-m^2+k*m*Exp[\[Nu]*x],y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^x-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+y(x)\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+y(x))}dK[2]+\int _1^{y(x)}\left (\frac {\exp \left (-\int _1^x-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{k \nu (m+K[3])^2}-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right ) \left (e^{\nu K[2]} k-m+K[3]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])^2}-\frac {\exp \left (-\int _1^{K[2]}-\frac {e^{\nu K[1]} k-2 m}{e^{\lambda K[1]} a+b e^{\mu K[1]}+c}dK[1]\right )}{\left (e^{\lambda K[2]} a+b e^{\mu K[2]}+c\right ) k \nu (m+K[3])}\right )dK[2]\right )dK[3]=c_1,y(x)\right ] \]