61.4.4 problem 25

Internal problem ID [12109]
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 : 25
Date solved : Tuesday, January 28, 2025 at 07:10:02 PM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 89

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

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

✓ Solution by Mathematica

Time used: 0.923 (sec). Leaf size: 254

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

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {e^{\frac {a x^{n+1}}{n+1}}}{\left (\lambda +e^{x \lambda } K[2]\right )^2}-\int _1^x\left (\frac {2 e^{\frac {a K[1]^{n+1}}{n+1}} \left (a \lambda K[1]^n+a e^{\lambda K[1]} K[2] K[1]^n+e^{2 \lambda K[1]} K[2]^2\right )}{\left (\lambda +e^{\lambda K[1]} K[2]\right )^3}-\frac {e^{\frac {a K[1]^{n+1}}{n+1}-\lambda K[1]} \left (a e^{\lambda K[1]} K[1]^n+2 e^{2 \lambda K[1]} K[2]\right )}{\left (\lambda +e^{\lambda K[1]} K[2]\right )^2}\right )dK[1]\right )dK[2]+\int _1^x-\frac {e^{\frac {a K[1]^{n+1}}{n+1}-\lambda K[1]} \left (a \lambda K[1]^n+a e^{\lambda K[1]} y(x) K[1]^n+e^{2 \lambda K[1]} y(x)^2\right )}{\left (\lambda +e^{\lambda K[1]} y(x)\right )^2}dK[1]=c_1,y(x)\right ] \]