61.4.8 problem 29

Internal problem ID [12113]
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 : 29
Date solved : Tuesday, January 28, 2025 at 12:25:04 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=a \,x^{n} y^{2}+\lambda y-a \,b^{2} x^{n} {\mathrm e}^{2 \lambda x} \end{align*}

✓ Solution by Maple

Time used: 0.031 (sec). Leaf size: 61

dsolve(diff(y(x),x)=a*x^n*y(x)^2+lambda*y(x)-a*b^2*x^n*exp(2*lambda*x),y(x), singsol=all)
 

\[ y = \tanh \left (\frac {-x^{n} a b \left (\Gamma \left (n , -\lambda x \right ) n -\Gamma \left (n +1\right )\right ) \left (-\lambda x \right )^{-n}-b a \,x^{n} {\mathrm e}^{\lambda x}+i \lambda c_{1}}{\lambda }\right ) b \,{\mathrm e}^{\lambda x} \]

✓ Solution by Mathematica

Time used: 0.998 (sec). Leaf size: 57

DSolve[D[y[x],x]==a*x^n*y[x]^2+\[Lambda]*y[x]-a*b^2*x^n*Exp[2*\[Lambda]*x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \sqrt {-b^2} e^{\lambda x} \tan \left (\frac {a \sqrt {-b^2} x^n (\lambda (-x))^{-n} \Gamma (n+1,-x \lambda )}{\lambda }+c_1\right ) \]