61.4.14 problem 35

Internal problem ID [12119]
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 : 35
Date solved : Tuesday, January 28, 2025 at 12:25:35 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.029 (sec). Leaf size: 38

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

\[ y = -\tan \left (x^{k} a b \left (\Gamma \left (k , -\lambda x \right )-\Gamma \left (k \right )\right ) \left (-\lambda x \right )^{-k}+c_{1} \right ) b \,x^{k} \]

✓ Solution by Mathematica

Time used: 0.939 (sec). Leaf size: 47

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

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