61.2.17 problem 17

Internal problem ID [12023]
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 : 17
Date solved : Monday, January 27, 2025 at 11:51:11 PM
CAS classification : [_rational, _Riccati]

\begin{align*} \left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 1131

dsolve((c__2*x^2+b__2*x+a__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0=0,y(x), singsol=all)
 

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✓ Solution by Mathematica

Time used: 4.533 (sec). Leaf size: 1045

DSolve[(c2*x^2+b2*x+a2)*(D[y[x],x]+\[Lambda]*y[x]^2)+a0==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (8 \text {c2} \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) G_{2,2}^{2,0}\left (-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}| \begin {array}{c} \frac {1}{4}-\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{4 \sqrt {\text {c2}}},\frac {1}{4} \left (\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}+1\right ) \\ 0,0 \\ \end {array} \right )+c_1 \left (8 \text {c2} \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )+\left (\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}-3 \text {c2}\right ) \left (\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}+3 \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (7-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right )^2 \left (G_{2,2}^{2,0}\left (-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}| \begin {array}{c} \frac {1}{4} \left (5-\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}\right ),\frac {1}{4} \left (\frac {\sqrt {\text {c2}-4 \text {a0} \lambda }}{\sqrt {\text {c2}}}+5\right ) \\ 0,1 \\ \end {array} \right )-\frac {4 \text {c2} c_1 (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )}{4 \text {a2} \text {c2}-\text {b2}^2}\right )} \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (2 \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )-(\text {a0} \lambda +2 \text {c2}) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (7-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )} \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (2 \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )-(\text {a0} \lambda +2 \text {c2}) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {7}{4}-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}+7\right ),3,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )\right )}{2 \lambda \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) (\text {a2}+x (\text {b2}+\text {c2} x)) \operatorname {Hypergeometric2F1}\left (\frac {3 \text {c2}+\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{4 \text {c2}},\frac {1}{4} \left (3-\frac {\sqrt {\text {c2} (\text {c2}-4 \text {a0} \lambda )}}{\text {c2}}\right ),2,-\frac {4 \text {c2} (\text {a2}+x (\text {b2}+\text {c2} x))}{\text {b2}^2-4 \text {a2} \text {c2}}\right )} \\ \end{align*}