61.4.1 problem 22

Internal problem ID [12106]
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 : 22
Date solved : Tuesday, January 28, 2025 at 12:24:33 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 85

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

\[ y = \frac {-c_{1} \lambda ^{2} x +\left (\int \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{x^{2}}d x \right ) x +{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{x^{2} \left (c_{1} \lambda ^{2}-\int \frac {{\mathrm e}^{\frac {a \,{\mathrm e}^{\lambda x} \left (\lambda x -1\right )}{\lambda ^{2}}}}{x^{2}}d x \right )} \]

✓ Solution by Mathematica

Time used: 1.034 (sec). Leaf size: 123

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

\begin{align*} y(x)\to -\frac {\exp \left (-\int _1^x-a e^{\lambda K[1]} K[1]dK[1]\right )+x \int _1^x\frac {\exp \left (-\int _1^{K[2]}-a e^{\lambda K[1]} K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1 x}{x^2 \left (\int _1^x\frac {\exp \left (-\int _1^{K[2]}-a e^{\lambda K[1]} K[1]dK[1]\right )}{K[2]^2}dK[2]+c_1\right )} \\ y(x)\to -\frac {1}{x} \\ \end{align*}