61.15.3 problem 12

Internal problem ID [12245]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-2. Equations containing arccosine.
Problem number : 12
Date solved : Tuesday, January 28, 2025 at 01:37:11 AM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=-\left (k +1\right ) x^{k} y^{2}+\lambda \arccos \left (x \right )^{n} \left (x^{k +1} y-1\right ) \end{align*}

✓ Solution by Maple

Time used: 0.015 (sec). Leaf size: 184

dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+lambda*arccos(x)^n*(x^(k+1)*y(x)-1),y(x), singsol=all)
 

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

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

DSolve[D[y[x],x]==-(k+1)*x^k*y[x]^2+\[Lambda]*ArcCos[x]^n*(x^(k+1)*y[x]-1),y[x],x,IncludeSingularSolutions -> True]
 

Not solved