Internal problem ID [9661]
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: 74.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} \left (a \,x^{n}-1\right ) \left (y^{\prime }+\lambda y^{2}\right )+\left (p \,x^{n}+q \right ) x y+r \,x^{n}+s=0} \end {gather*}
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 3726
dsolve(x^2*(a*x^n-1)*(diff(y(x),x)+lambda*y(x)^2)+(p*x^n+q)*x*y(x)+r*x^n+s=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 8.555 (sec). Leaf size: 2419
DSolve[x^2*(a*x^n-1)*(y'[x]+\[Lambda]*y[x]^2)+(p*x^n+q)*x*y[x]+r*x^n+s==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {\left (q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}\right ) \left (-q a+\sqrt {q^2+2 q+4 s \lambda +1} a-p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}\right ) c_1 \, _2F_1\left (\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};2-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right ) x^n}{a \left (\sqrt {q^2+2 q+4 s \lambda +1}-n\right )}+2 \left (q-\sqrt {q^2+2 q+4 s \lambda +1}+1\right ) c_1 \, _2F_1\left (\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};1-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )+2 i^{\frac {2 \sqrt {q^2+2 q+4 s \lambda +1}}{n}} a^{\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}} \left (x^n\right )^{\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}} \left (q+\sqrt {q^2+2 q+4 s \lambda +1}+1\right ) \, _2F_1\left (\frac {p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};\frac {n+\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )+\frac {i^{\frac {2 \sqrt {q^2+2 q+4 s \lambda +1}}{n}} a^{\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}-1} \left (x^n\right )^{\frac {n+\sqrt {q^2+2 q+4 s \lambda +1}}{n}} \left (p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}\right ) \left (p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}\right ) \, _2F_1\left (\frac {p+a \left (2 n+q+\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (2 n+q+\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}+2;a x^n\right )}{n+\sqrt {q^2+2 q+4 s \lambda +1}}}{4 x \lambda \left (i^{\frac {2 \sqrt {q^2+2 q+4 s \lambda +1}}{n}} a^{\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}} \, _2F_1\left (\frac {p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (q+\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};\frac {n+\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right ) \left (x^n\right )^{\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n}}+c_1 \, _2F_1\left (\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};1-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )\right )} \\ y(x)\to \frac {-\frac {x^n \left (a \left (q^2 \left (n+\sqrt {q^2+2 q+4 \lambda s+1}-2\right )+q \left (n \left (-\sqrt {q^2+2 q+4 \lambda s+1}\right )+n+\sqrt {q^2+2 q+4 \lambda s+1}-4 \lambda s-1\right )+2 \lambda s \left (n+\sqrt {q^2+2 q+4 \lambda s+1}\right )-q^3\right )+p \left (n \left (-\sqrt {q^2+2 q+4 \lambda s+1}+q+1\right )+q \left (\sqrt {q^2+2 q+4 \lambda s+1}-2\right )+\sqrt {q^2+2 q+4 \lambda s+1}-q^2-4 \lambda s-1\right )+2 \lambda r \left (n+\sqrt {q^2+2 q+4 \lambda s+1}\right )\right ) \, _2F_1\left (\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};2-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )}{\left (-n^2+q^2+2 q+4 \lambda s+1\right ) \, _2F_1\left (\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};1-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )}-\sqrt {q^2+2 q+4 \lambda s+1}+q+1}{2 \lambda x} \\ y(x)\to \frac {-\frac {x^n \left (a \left (q^2 \left (n+\sqrt {q^2+2 q+4 \lambda s+1}-2\right )+q \left (n \left (-\sqrt {q^2+2 q+4 \lambda s+1}\right )+n+\sqrt {q^2+2 q+4 \lambda s+1}-4 \lambda s-1\right )+2 \lambda s \left (n+\sqrt {q^2+2 q+4 \lambda s+1}\right )-q^3\right )+p \left (n \left (-\sqrt {q^2+2 q+4 \lambda s+1}+q+1\right )+q \left (\sqrt {q^2+2 q+4 \lambda s+1}-2\right )+\sqrt {q^2+2 q+4 \lambda s+1}-q^2-4 \lambda s-1\right )+2 \lambda r \left (n+\sqrt {q^2+2 q+4 \lambda s+1}\right )\right ) \, _2F_1\left (\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {p+a \left (2 n+q-\sqrt {q^2+2 q+4 s \lambda +1}\right )+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};2-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )}{\left (-n^2+q^2+2 q+4 \lambda s+1\right ) \, _2F_1\left (\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p-\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n},\frac {q a-\sqrt {q^2+2 q+4 s \lambda +1} a+p+\sqrt {a^2-2 (p+2 r \lambda ) a+p^2}}{2 a n};1-\frac {\sqrt {q^2+2 q+4 s \lambda +1}}{n};a x^n\right )}-\sqrt {q^2+2 q+4 \lambda s+1}+q+1}{2 \lambda x} \\ \end{align*}