61.2.74 problem 74

Internal problem ID [12080]
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
Date solved : Tuesday, January 28, 2025 at 12:15:14 AM
CAS classification : [_rational, _Riccati]

\begin{align*} 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{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 1222

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: 4.584 (sec). Leaf size: 1882

DSolve[x^2*(a*x^n-1)*(D[y[x],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 \operatorname {Hypergeometric2F1}\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 \operatorname {Hypergeometric2F1}\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 ) \operatorname {Hypergeometric2F1}\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 ) \operatorname {Hypergeometric2F1}\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}} \operatorname {Hypergeometric2F1}\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 \operatorname {Hypergeometric2F1}\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 ) \operatorname {Hypergeometric2F1}\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 ) \operatorname {Hypergeometric2F1}\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*}