61.4.12 problem 33

Internal problem ID [12117]
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 : 33
Date solved : Tuesday, January 28, 2025 at 12:25:25 AM
CAS classification : [_Riccati]

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 114

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

\[ y = \frac {\left (\tan \left (\frac {\sqrt {4 a \,b^{2} c -b^{4}}\, \left (x^{n} \left (-\lambda x \right )^{-n} \Gamma \left (n , -\lambda x \right ) b n -x^{n} \left (-\lambda x \right )^{-n} \Gamma \left (n +1\right ) b +x^{n} {\mathrm e}^{\lambda x} b +\lambda c_{1} \right )}{2 b^{2} \lambda }\right ) \sqrt {4 a \,b^{2} c -b^{4}}-b^{2}\right ) {\mathrm e}^{-\lambda x}}{2 a b} \]

✓ Solution by Mathematica

Time used: 1.486 (sec). Leaf size: 102

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

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