61.2.78 problem 78

Internal problem ID [12084]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number : 78
Date solved : Tuesday, January 28, 2025 at 12:20:43 AM
CAS classification : [[_homogeneous, `class D`], _rational, _Riccati]

\begin{align*} \left (a \,x^{n}+b \,x^{m}+c \right ) \left (-y+x y^{\prime }\right )+s \,x^{k} \left (y^{2}-\lambda \,x^{2}\right )&=0 \end{align*}

✓ Solution by Maple

Time used: 0.087 (sec). Leaf size: 37

dsolve((a*x^n+b*x^m+c)*(x*diff(y(x),x)-y(x))+s*x^k*(y(x)^2-lambda*x^2)=0,y(x), singsol=all)
 

\[ y = \tanh \left (s \sqrt {\lambda }\, \left (\int \frac {x^{k}}{a \,x^{n}+x^{m} b +c}d x +c_{1} \right )\right ) x \sqrt {\lambda } \]

✓ Solution by Mathematica

Time used: 0.750 (sec). Leaf size: 59

DSolve[(a*x^n+b*x^m+c)*(x*D[y[x],x]-y[x])+s*x^k*(y[x]^2-\[Lambda]*x^2)==0,y[x],x,IncludeSingularSolutions -> True]
 

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