Internal problem ID [9614]
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: 27.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-\left (\alpha x +\beta \right ) y-a \,x^{2}-x b -c=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 4562
dsolve(diff(y(x),x)=y(x)^2+(alpha*x+beta)*y(x)+a*x^2+b*x+c,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.195 (sec). Leaf size: 1291
DSolve[y'[x]==y[x]^2+(\[Alpha]*x+\[Beta])*y[x]+a*x^2+b*x+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \left (2 b+4 a x+\left (\sqrt {\alpha ^2-4 a}-\alpha \right ) (x \alpha +\beta )\right ) \, _1F_1\left (-\frac {2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -\sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )}{4 \left (\alpha ^2-4 a\right )^{3/2}};\frac {1}{2};\frac {(2 b+4 a x-\alpha (x \alpha +\beta ))^2}{2 \left (\alpha ^2-4 a\right )^{3/2}}\right ) \left (\alpha ^2-4 a\right )^2+2 (2 b+4 a x-\alpha (x \alpha +\beta )) \left (2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -\sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )\right ) \, _1F_1\left (-\frac {2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -5 \sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +10 \sqrt {\alpha ^2-4 a}\right )}{4 \left (\alpha ^2-4 a\right )^{3/2}};\frac {3}{2};\frac {(2 b+4 a x-\alpha (x \alpha +\beta ))^2}{2 \left (\alpha ^2-4 a\right )^{3/2}}\right ) \sqrt {\alpha ^2-4 a}+\left (4 a-\alpha ^2\right ) c_1 \left (2 \left (4 a-\alpha ^2\right ) \left (2 b+4 a x+\left (\sqrt {\alpha ^2-4 a}-\alpha \right ) (x \alpha +\beta )\right ) H_{-\frac {-2 b^2+2 \alpha \beta b+\alpha ^2 \left (-2 c-\alpha +\sqrt {\alpha ^2-4 a}\right )-2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )}{2 \left (\alpha ^2-4 a\right )^{3/2}}}\left (\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )-2 \sqrt {2} \sqrt [4]{\alpha ^2-4 a} \left (2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -\sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )\right ) H_{\frac {2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -3 \sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +6 \sqrt {\alpha ^2-4 a}\right )}{2 \left (\alpha ^2-4 a\right )^{3/2}}}\left (\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )\right )}{4 \left (\alpha ^2-4 a\right )^{5/2} \left (c_1 H_{-\frac {-2 b^2+2 \alpha \beta b+\alpha ^2 \left (-2 c-\alpha +\sqrt {\alpha ^2-4 a}\right )-2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )}{2 \left (\alpha ^2-4 a\right )^{3/2}}}\left (\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )+\, _1F_1\left (-\frac {2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -\sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )}{4 \left (\alpha ^2-4 a\right )^{3/2}};\frac {1}{2};\frac {(2 b+4 a x-\alpha (x \alpha +\beta ))^2}{2 \left (\alpha ^2-4 a\right )^{3/2}}\right )\right )} \\ y(x)\to \frac {\left (4 a-\alpha ^2\right ) \left (\left (\sqrt {\alpha ^2-4 a}-\alpha \right ) (\beta +\alpha x)+4 a x+2 b\right )-\frac {\sqrt {2} \sqrt [4]{\alpha ^2-4 a} \left (2 a \left (2 \sqrt {\alpha ^2-4 a}-2 \alpha +\beta ^2-4 c\right )+\alpha ^2 \left (-\sqrt {\alpha ^2-4 a}+\alpha +2 c\right )+2 b^2-2 \alpha \beta b\right ) H_{\frac {2 b^2-2 \alpha \beta b+\alpha ^2 \left (2 c+\alpha -3 \sqrt {\alpha ^2-4 a}\right )+2 a \left (\beta ^2-4 c-2 \alpha +6 \sqrt {\alpha ^2-4 a}\right )}{2 \left (\alpha ^2-4 a\right )^{3/2}}}\left (\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )}{H_{-\frac {-2 b^2+2 \alpha \beta b+\alpha ^2 \left (-2 c-\alpha +\sqrt {\alpha ^2-4 a}\right )-2 a \left (\beta ^2-4 c-2 \alpha +2 \sqrt {\alpha ^2-4 a}\right )}{2 \left (\alpha ^2-4 a\right )^{3/2}}}\left (\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )}}{2 \left (\alpha ^2-4 a\right )^{3/2}} \\ \end{align*}