Internal problem ID [9604]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (c_{2} x^{2}+b_{2} x +a_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 3288
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)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 1.892 (sec). Leaf size: 846
DSolve[(c2*x^2+b2*x+a2)*(y'[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}}| {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 \\ \\ \right )+4 \text {c2} c_1 \left (2 \left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \, _2F_1\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)) \, _2F_1\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}}| {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 \\ \\ \right )+\frac {4 \text {c2} c_1 (\text {a2}+x (\text {b2}+\text {c2} x)) \, _2F_1\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 {b2}^2-4 \text {a2} \text {c2}}\right )} \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (\frac {2}{\text {a2}+x (\text {b2}+\text {c2} x)}-\frac {(\text {a0} \lambda +2 \text {c2}) \, _2F_1\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 )}{\left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \, _2F_1\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 )}\right )}{2 \lambda } \\ y(x)\to \frac {(\text {b2}+2 \text {c2} x) \left (\frac {2}{\text {a2}+x (\text {b2}+\text {c2} x)}-\frac {(\text {a0} \lambda +2 \text {c2}) \, _2F_1\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 )}{\left (\text {b2}^2-4 \text {a2} \text {c2}\right ) \, _2F_1\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 )}\right )}{2 \lambda } \\ \end{align*}