Internal problem ID [9692]
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]
\[ \boxed {y^{\prime }+\left (1+k \right ) x^{k} y^{2}-a \,x^{1+k} {\mathrm e}^{\lambda x} y+{\mathrm e}^{\lambda x} a=0} \]
✓ 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: 0.0 (sec). Leaf size: 0
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]
Not solved