61.24.9 problem 9

Internal problem ID [12422]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 07:58:32 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 1972

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

\[ \text {Expression too large to display} \]

✓ Solution by Mathematica

Time used: 0.184 (sec). Leaf size: 113

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

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