Internal problem ID [9831]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. subsection 1.2.7-3. Equations containing
arctangent.
Problem number: 21.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }+\left (1+k \right ) x^{k} y^{2}-\lambda \arctan \left (x \right )^{n} \left (x^{1+k} y-1\right )=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 205
dsolve(diff(y(x),x)=-(k+1)*x^k*y(x)^2+lambda*arctan(x)^n*(x^(k+1)*y(x)-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left ({\mathrm e}^{\int \frac {x^{k} \arctan \left (x \right )^{n} \lambda \,x^{2}-2 k -2}{x}d x} x^{k} x -\left (\int \left (-x^{k} k \,{\mathrm e}^{\lambda \left (\int x^{1+k} \arctan \left (x \right )^{n}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{k} {\mathrm e}^{\lambda \left (\int x^{1+k} \arctan \left (x \right )^{n}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x \right )-c_{1} \right ) x^{-k}}{x \left (\int \left (-x^{k} k \,{\mathrm e}^{\lambda \left (\int x^{1+k} \arctan \left (x \right )^{n}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}-x^{k} {\mathrm e}^{\lambda \left (\int x^{1+k} \arctan \left (x \right )^{n}d x \right )-2 k \left (\int \frac {1}{x}d x \right )-2 \left (\int \frac {1}{x}d x \right )}\right )d x +c_{1} \right )} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==-(k+1)*x^k*y[x]^2+\[Lambda]*ArcTan[x]^n*(x^(k+1)*y[x]-1),y[x],x,IncludeSingularSolutions -> True]
Not solved