Internal problem ID [9625]
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: 38.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.022 (sec). Leaf size: 848
dsolve(x*diff(y(x),x)+a__3*x*y(x)^2+a__2*y(x)+a__1*x+a__0=0,y(x), singsol=all)
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✓ Solution by Mathematica
Time used: 0.709 (sec). Leaf size: 541
DSolve[x*y'[x]+a3*x*y[x]^2+a2*y[x]+a1*x+a0==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \left (\sqrt {\text {a1}} c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (\text {a2}+\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+c_1 \left (\sqrt {\text {a1}} \text {a2}+i \text {a0} \sqrt {\text {a3}}\right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+\sqrt {\text {a1}} \left (2 L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}-1}^{\text {a2}}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )\right )}{\sqrt {\text {a3}} \left (c_1 \text {HypergeometricU}\left (\frac {1}{2} \left (\text {a2}+\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )+L_{-\frac {i \sqrt {\text {a3}} \text {a0}}{2 \sqrt {\text {a1}}}-\frac {\text {a2}}{2}}^{\text {a2}-1}\left (2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )\right )} \\ y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\text {HypergeometricU}\left (\frac {1}{2} \left (\text {a2}+\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\ y(x)\to \frac {\frac {\left (\text {a0} \sqrt {\text {a3}}-i \sqrt {\text {a1}} \text {a2}\right ) \text {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\text {HypergeometricU}\left (\frac {1}{2} \left (\text {a2}+\frac {i \text {a0} \sqrt {\text {a3}}}{\sqrt {\text {a1}}}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \\ \end{align*}