61.4.13 problem 34

Internal problem ID [12118]
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 : 34
Date solved : Tuesday, January 28, 2025 at 12:25:30 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime }&=a \,{\mathrm e}^{\lambda x} \left (y-b \,x^{n}-c \right )^{2}+b n \,x^{n -1} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 50

dsolve(diff(y(x),x)=a*exp(lambda*x)*(y(x)-b*x^n-c)^2+b*n*x^(n-1),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 1.541 (sec). Leaf size: 40

DSolve[D[y[x],x]==a*Exp[\[Lambda]*x]*(y[x]-b*x^n-c)^2+b*n*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {\lambda }{-a e^{\lambda x}+c_1 \lambda }+b x^n+c \\ y(x)\to b x^n+c \\ \end{align*}