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 (x^{n} a -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: 0.0 (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: 4.971 (sec). Leaf size: 1619
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 {a^{\frac {n+\sqrt {(q+1)^2+4 s \lambda }}{n}} e^{\frac {i \pi \sqrt {(q+1)^2+4 s \lambda }}{n}} \left (\frac {2 n \left (2 r \lambda +2 a s \lambda +p \left (q+\sqrt {(q+1)^2+4 s \lambda }+1\right )+a q \left (q+\sqrt {(q+1)^2+4 s \lambda }+1\right )\right ) \, _2F_1\left (\frac {p+a \left (2 n+q+\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (2 n+q+\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {\sqrt {(q+1)^2+4 s \lambda }}{n}+2;a x^n\right ) x^n}{n+\sqrt {(q+1)^2+4 s \lambda }}+2 n \left (q+\sqrt {(q+1)^2+4 s \lambda }+1\right ) \, _2F_1\left (\frac {p+a \left (q+\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (q+\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {n+\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right )\right ) \left (x^n\right )^{\frac {\sqrt {(q+1)^2+4 s \lambda }}{n}}+c_1 \left (\frac {2 a n \left (a \sqrt {(q+1)^2+4 s \lambda } q-2 r \lambda -a \left (q^2+q+2 s \lambda \right )+p \left (-q+\sqrt {(q+1)^2+4 s \lambda }-1\right )\right ) \, _2F_1\left (\frac {p+a \left (2 n+q-\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (2 n+q-\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};2-\frac {\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right ) x^n}{\sqrt {(q+1)^2+4 s \lambda }-n}+2 a n \left (q-\sqrt {(q+1)^2+4 s \lambda }+1\right ) \, _2F_1\left (\frac {q a-\sqrt {(q+1)^2+4 s \lambda } a+p-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {q a-\sqrt {(q+1)^2+4 s \lambda } a+p+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {n-\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right )\right )}{4 a n x \lambda \left (a^{\frac {\sqrt {(q+1)^2+4 s \lambda }}{n}} e^{\frac {i \pi \sqrt {(q+1)^2+4 s \lambda }}{n}} \, _2F_1\left (\frac {p+a \left (q+\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (q+\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {n+\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right ) \left (x^n\right )^{\frac {\sqrt {(q+1)^2+4 s \lambda }}{n}}+c_1 \, _2F_1\left (\frac {q a-\sqrt {(q+1)^2+4 s \lambda } a+p-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {q a-\sqrt {(q+1)^2+4 s \lambda } a+p+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {n-\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right )\right )} \\ y(x)\to \frac {\frac {x^n \left (a \left (q^2 \left (n+\sqrt {(q+1)^2+4 \lambda s}-2\right )+q \left (n \left (-\sqrt {(q+1)^2+4 \lambda s}\right )+n+\sqrt {(q+1)^2+4 \lambda s}-4 \lambda s-1\right )+2 \lambda s \left (n+\sqrt {(q+1)^2+4 \lambda s}\right )-q^3\right )+p \left (n \left (-\sqrt {(q+1)^2+4 \lambda s}+q+1\right )+q \left (\sqrt {(q+1)^2+4 \lambda s}-q-2\right )+\sqrt {(q+1)^2+4 \lambda s}-4 \lambda s-1\right )+2 \lambda r \left (n+\sqrt {(q+1)^2+4 \lambda s}\right )\right ) \, _2F_1\left (\frac {p+a \left (2 n+q-\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (2 n+q-\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};2-\frac {\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right )}{\left (n^2-(q+1)^2-4 \lambda s\right ) \, _2F_1\left (\frac {p+a \left (q-\sqrt {(q+1)^2+4 s \lambda }\right )-\sqrt {(a-p)^2-4 a r \lambda }}{2 a n},\frac {p+a \left (q-\sqrt {(q+1)^2+4 s \lambda }\right )+\sqrt {(a-p)^2-4 a r \lambda }}{2 a n};\frac {n-\sqrt {(q+1)^2+4 s \lambda }}{n};a x^n\right )}-\sqrt {(q+1)^2+4 \lambda s}+q+1}{2 \lambda x} \\ \end{align*}