Internal problem ID [9624]
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: 37.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x -y^{2} x -a y-b \,x^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 216
dsolve(x*diff(y(x),x)=x*y(x)^2+a*y(x)+b*x^n,y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} x^{\frac {n}{2}+\frac {1}{2}} \sqrt {b}\, \BesselY \left (-\frac {a -n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )}{\left (\BesselY \left (-\frac {a +1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1}+\BesselJ \left (-\frac {a +1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )\right ) x}+\frac {\BesselJ \left (-\frac {a -n}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{\left (\BesselY \left (-\frac {a +1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right ) c_{1}+\BesselJ \left (-\frac {a +1}{n +1}, \frac {2 \sqrt {b}\, x^{\frac {n}{2}+\frac {1}{2}}}{n +1}\right )\right ) x} \]
✓ Solution by Mathematica
Time used: 0.711 (sec). Leaf size: 650
DSolve[x*y'[x]==x*y[x]^2+a*y[x]+b*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 (a+1) \Gamma \left (\frac {a+1}{n+1}\right ) \left ((a+1) J_{\frac {a+1}{n+1}}\left (\frac {2 \sqrt {b} \left (x^n\right )^{\frac {1}{2} \left (1+\frac {1}{n}\right )}}{n+1}\right )-\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}} J_{\frac {a+n+2}{n+1}}\left (\frac {2 \sqrt {b} \left (x^n\right )^{\frac {1}{2} \left (1+\frac {1}{n}\right )}}{n+1}\right )\right )+c_1 \Gamma \left (\frac {n-a}{n+1}\right ) \left (\frac {\sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )^{-\frac {a+1}{n+1}} \left (-b \left (x^n\right )^{\frac {1}{n}+1} \, _0\tilde {F}_1\left (;\frac {n-a}{n+1}+1;-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )+(n+1)^2 \, _0\tilde {F}_1\left (;-\frac {a+1}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )+(a+1) (n+1) \, _0\tilde {F}_1\left (;\frac {n-a}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )\right )}{2 (n+1) x \left (\Gamma \left (\frac {a+n+2}{n+1}\right ) J_{\frac {a+1}{n+1}}\left (\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )+c_1 \Gamma \left (\frac {n-a}{n+1}\right ) J_{-\frac {a+1}{n+1}}\left (\frac {2 \sqrt {b} \left (x^n\right )^{\frac {n+1}{2 n}}}{n+1}\right )\right )} \\ y(x)\to -\frac {\frac {(n+1) \left (\, _0\tilde {F}_1\left (;-\frac {a+1}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )-\frac {b \left (x^n\right )^{\frac {1}{n}+1} \, _0\tilde {F}_1\left (;\frac {n-a}{n+1}+1;-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )}{(n+1)^2}\right )}{\, _0\tilde {F}_1\left (;\frac {n-a}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )}+a+1}{2 x} \\ y(x)\to -\frac {\frac {(n+1) \left (\, _0\tilde {F}_1\left (;-\frac {a+1}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )-\frac {b \left (x^n\right )^{\frac {1}{n}+1} \, _0\tilde {F}_1\left (;\frac {n-a}{n+1}+1;-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )}{(n+1)^2}\right )}{\, _0\tilde {F}_1\left (;\frac {n-a}{n+1};-\frac {b \left (x^n\right )^{1+\frac {1}{n}}}{(n+1)^2}\right )}+a+1}{2 x} \\ \end{align*}