61.2.27 problem 27

Internal problem ID [12033]
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
Date solved : Monday, January 27, 2025 at 11:51:54 PM
CAS classification : [_Riccati]

\begin{align*} y^{\prime }&=y^{2}+\left (\alpha x +\beta \right ) y+a \,x^{2}+b x +c \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 974

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: 2.446 (sec). Leaf size: 1640

DSolve[D[y[x],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 {\left (2 b+4 a x+\left (\sqrt {\alpha ^2-4 a}-\alpha \right ) (x \alpha +\beta )\right ) \operatorname {Hypergeometric1F1}\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 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 ) \operatorname {Hypergeometric1F1}\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 (\left (4 a-\alpha ^2\right ) \left (2 b+4 a x+\left (\sqrt {\alpha ^2-4 a}-\alpha \right ) (x \alpha +\beta )\right ) \operatorname {HermiteH}\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 )}{2 \left (\alpha ^2-4 a\right )^{3/2}},\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )-\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 ) \operatorname {HermiteH}\left (\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}},\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )\right )}{2 \left (\alpha ^2-4 a\right )^{5/2} \left (c_1 \operatorname {HermiteH}\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 )}{2 \left (\alpha ^2-4 a\right )^{3/2}},\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )+\operatorname {Hypergeometric1F1}\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 ) \operatorname {HermiteH}\left (\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}},\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )}{\operatorname {HermiteH}\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 )}{2 \left (\alpha ^2-4 a\right )^{3/2}},\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}} \\ 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 ) \operatorname {HermiteH}\left (\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}},\frac {-2 b-4 a x+\alpha (x \alpha +\beta )}{\sqrt {2} \left (\alpha ^2-4 a\right )^{3/4}}\right )}{\operatorname {HermiteH}\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 )}{2 \left (\alpha ^2-4 a\right )^{3/2}},\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*}