Internal problem ID [10449]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3-2. Equations with power and
exponential functions
Problem number: 31.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (k +1\right ) x^{k} y^{2}-a \,x^{k +1} {\mathrm e}^{\lambda x} y=-{\mathrm e}^{\lambda x} a} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 196
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+a*x^(k+1)*exp(lambda*x)*y(x)-a*exp(lambda*x),y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left ({\mathrm e}^{\int \frac {x^{2} x^{k} {\mathrm e}^{\lambda x} a -2 k -2}{x}d x} x \,x^{k}+\int \left (x^{k} k \,{\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{\lambda x}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}+x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{\lambda x}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right ) x^{-k}}{x \left (\int \left (x^{k} k \,{\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{\lambda x}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}+x^{k} {\mathrm e}^{a \left (\int x^{1+k} {\mathrm e}^{\lambda x}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right )} \]
✓ Solution by Mathematica
Time used: 86.249 (sec). Leaf size: 280
DSolve[y'[x]==-(k+1)*x^k*y[x]^2+a*x^(k+1)*Exp[\[Lambda]*x]*y[x]-a*Exp[\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {a \lambda \exp \left (\frac {a \left (-\log \left (e^{\lambda x}\right )\right )^{-k} \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^k \Gamma \left (k+2,-\log \left (e^{x \lambda }\right )\right )}{\lambda ^2}\right ) \left (1+c_1 \int _1^{e^{x \lambda }}\exp \left (-\frac {a \Gamma (k+2,-\log (K[1])) (-\log (K[1]))^{-k} \left (\frac {\log (K[1])}{\lambda }\right )^k}{\lambda ^2}\right )dK[1]\right )}{a c_1 \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\frac {a \left (-\log \left (e^{\lambda x}\right )\right )^{-k} \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^k \Gamma \left (k+2,-\log \left (e^{x \lambda }\right )\right )}{\lambda ^2}\right ) \int _1^{e^{x \lambda }}\exp \left (-\frac {a \Gamma (k+2,-\log (K[1])) (-\log (K[1]))^{-k} \left (\frac {\log (K[1])}{\lambda }\right )^k}{\lambda ^2}\right )dK[1]+a \lambda \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^{k+1} \exp \left (\frac {a \left (-\log \left (e^{\lambda x}\right )\right )^{-k} \left (\frac {\log \left (e^{\lambda x}\right )}{\lambda }\right )^k \Gamma \left (k+2,-\log \left (e^{x \lambda }\right )\right )}{\lambda ^2}\right )-c_1 \lambda ^2} \]