Internal problem ID [9599]
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: 12.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {\left (a_{2} x +b_{2} \right ) \left (y^{\prime }+\lambda y^{2}\right )+a_{0} x +b_{0}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 890
dsolve((a__2*x+b__2)*(diff(y(x),x)+lambda*y(x)^2)+a__0*x+b__0=0,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 0.803 (sec). Leaf size: 471
DSolve[(a2*x+b2)*(y'[x]+\[Lambda]*y[x]^2)+a0*x+b0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\frac {i \text {a2}^{3/2} c_1 \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2}) \text {HypergeometricU}\left (1+\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},1,\frac {2 i \sqrt {\text {a0}} \sqrt {\lambda } (\text {a2} x+\text {b2})}{\text {a2}^{3/2}}\right )+2 \sqrt {\text {a0}} \text {a2}^3 L_{\frac {i (\text {a0} \text {b2}-\text {a2} \text {b0}) \sqrt {\lambda }}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}-1}\left (\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}{i \text {a2}^{3/2} c_1 \text {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} \sqrt {\lambda } (\text {a2} x+\text {b2})}{\text {a2}^{3/2}}\right )+2 \sqrt {\text {a0}} \sqrt {\lambda } (\text {a2} x+\text {b2}) \, _1F_1\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}}+1;2;\frac {2 i \sqrt {\text {a0}} (\text {b2}+\text {a2} x) \sqrt {\lambda }}{\text {a2}^{3/2}}\right )}-i \sqrt {\text {a0}} \text {a2}^{3/2}}{\text {a2}^2 \sqrt {\lambda }} \\ y(x)\to \frac {(\text {a2} \text {b0}-\text {a0} \text {b2}) \text {HypergeometricU}\left (1+\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},1,\frac {2 i \sqrt {\text {a0}} \sqrt {\lambda } (\text {a2} x+\text {b2})}{\text {a2}^{3/2}}\right )}{\text {a2}^2 \text {HypergeometricU}\left (\frac {i \sqrt {\lambda } (\text {a2} \text {b0}-\text {a0} \text {b2})}{2 \sqrt {\text {a0}} \text {a2}^{3/2}},0,\frac {2 i \sqrt {\text {a0}} \sqrt {\lambda } (\text {a2} x+\text {b2})}{\text {a2}^{3/2}}\right )}-\frac {i \sqrt {\text {a0}}}{\sqrt {\text {a2}} \sqrt {\lambda }} \\ \end{align*}