2.2.35 Problem 38
Internal
problem
ID
[13241]
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
Date
solved
:
Wednesday, December 31, 2025 at 12:21:43 PM
CAS
classification
:
[_rational, _Riccati]
2.2.35.1 Solved using first_order_ode_riccati
15.460 (sec)
Entering first order ode riccati solver
\begin{align*}
y^{\prime } x +a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}&=0 \\
\end{align*}
In canonical form the ODE is \begin{align*} y' &= F(x,y)\\ &= -\frac {a_{3} x y^{2}+a_{2} y+a_{1} x +a_{0}}{x} \end{align*}
This is a Riccati ODE. Comparing the ODE to solve
\[
y' = \textit {the\_rhs}
\]
With Riccati ODE standard form \[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]
Shows
that \(f_0(x)=-a_{1} -\frac {a_{0}}{x}\), \(f_1(x)=-\frac {a_{2}}{x}\) and \(f_2(x)=-a_{3}\). Let \begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{-u a_{3}} \tag {1} \end{align*}
Using the above substitution in the given ODE results (after some simplification) in a second
order ODE to solve for \(u(x)\) which is
\begin{align*} f_2 u''(x) -\left ( f_2' + f_1 f_2 \right ) u'(x) + f_2^2 f_0 u(x) &= 0 \tag {2} \end{align*}
But
\begin{align*} f_2' &=0\\ f_1 f_2 &=\frac {a_{2} a_{3}}{x}\\ f_2^2 f_0 &=a_{3}^{2} \left (-a_{1} -\frac {a_{0}}{x}\right ) \end{align*}
Substituting the above terms back in equation (2) gives
\[
-a_{3} u^{\prime \prime }\left (x \right )-\frac {a_{2} a_{3} u^{\prime }\left (x \right )}{x}+a_{3}^{2} \left (-a_{1} -\frac {a_{0}}{x}\right ) u \left (x \right ) = 0
\]
Unable to solve. Will ask Maple to solve
this ode now.
The solution for \(u \left (x \right )\) is
\begin{equation}
\tag{3} u \left (x \right ) = c_1 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+c_2 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )
\end{equation}
Taking derivative gives \begin{equation}
\tag{4} u^{\prime }\left (x \right ) = -i c_1 \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {c_1 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\left (-\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}+a_{2} \right ) \operatorname {KummerM}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}-i c_2 \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {c_2 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )-\operatorname {KummerU}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}
\end{equation}
Substituting equations (3,4) into (1) results in \begin{align*}
y &= \frac {-u'}{f_2 u} \\
y &= \frac {-u'}{-u a_{3}} \\
y &= \frac {-i c_1 \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {c_1 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\left (-\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}+a_{2} \right ) \operatorname {KummerM}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}-i c_2 \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {c_2 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )-\operatorname {KummerU}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}}{a_{3} \left (c_1 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+c_2 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \\
\end{align*}
Doing change of constants, the above solution becomes \[
y = \frac {-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\left (-\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}+a_{2} \right ) \operatorname {KummerM}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}-i c_3 \sqrt {a_{1}}\, \sqrt {a_{3}}\, {\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+\frac {c_3 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \left (\left (2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x +\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}-a_{2} \right ) \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )-\operatorname {KummerU}\left (-1+\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{x}}{a_{3} \left ({\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )+c_3 \,{\mathrm e}^{-i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x} \operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}
\]
Simplifying the above gives
\begin{align*}
y &= \frac {\frac {\left (\left (-\frac {1}{2} a_{2}^{2}+a_{2} \right ) a_{1}^{{3}/{2}}+\left (i a_{1} \sqrt {a_{3}}-\frac {\sqrt {a_{1}}\, a_{3} a_{0}}{2}\right ) a_{0} \right ) c_3 \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i \sqrt {a_{3}}\, a_{0}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {\operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i \sqrt {a_{3}}\, a_{0}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) \left (i \sqrt {a_{3}}\, a_{1} a_{0} +a_{1}^{{3}/{2}} a_{2} \right )}{2}-\left (\frac {a_{1}^{{3}/{2}} a_{2}}{2}+i \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \sqrt {a_{3}}\right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_3 +\operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{{3}/{2}} x a_{3} \left (\operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_3 +\operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \\
\end{align*}
Summary of solutions found
\begin{align*}
y &= \frac {\frac {\left (\left (-\frac {1}{2} a_{2}^{2}+a_{2} \right ) a_{1}^{{3}/{2}}+\left (i a_{1} \sqrt {a_{3}}-\frac {\sqrt {a_{1}}\, a_{3} a_{0}}{2}\right ) a_{0} \right ) c_3 \operatorname {KummerU}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i \sqrt {a_{3}}\, a_{0}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )}{2}+\frac {\operatorname {KummerM}\left (\frac {\left (a_{2} +2\right ) \sqrt {a_{1}}+i \sqrt {a_{3}}\, a_{0}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) \left (i \sqrt {a_{3}}\, a_{1} a_{0} +a_{1}^{{3}/{2}} a_{2} \right )}{2}-\left (\frac {a_{1}^{{3}/{2}} a_{2}}{2}+i \left (a_{1} x +\frac {a_{0}}{2}\right ) a_{1} \sqrt {a_{3}}\right ) \left (\operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_3 +\operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )}{a_{1}^{{3}/{2}} x a_{3} \left (\operatorname {KummerU}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right ) c_3 +\operatorname {KummerM}\left (\frac {i \sqrt {a_{3}}\, a_{0} +a_{2} \sqrt {a_{1}}}{2 \sqrt {a_{1}}}, a_{2} , 2 i \sqrt {a_{1}}\, \sqrt {a_{3}}\, x \right )\right )} \\
\end{align*}
2.2.35.2 ✓ Maple. Time used: 0.016 (sec). Leaf size: 403
ode:=x*diff(y(x),x)+a__3*x*y(x)^2+a__2*y(x)+a__1*x+a__0 = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {4 a_{1} \left (a_{1}^{3} a_{3} \left (a_{3} a_{0} -a_{2} \sqrt {-a_{1} a_{3}}\right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\frac {c_1 \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )}{4}+a_{1}^{3} a_{3} \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\frac {\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_1 \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right )}{2}\right )}{4 a_{1}^{3} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )-\sqrt {-a_{1} a_{3}}\, c_1 \left (a_{0}^{2} a_{3} +a_{1} a_{2}^{2}\right ) \operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} \left (a_{2} +2\right )}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+2 a_{1} \left (-2 a_{1}^{2} a_{3}^{2} \left (\sqrt {-a_{1} a_{3}}\, a_{0} -a_{1} a_{2} \right ) \operatorname {KummerM}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right )+\operatorname {KummerU}\left (\frac {\sqrt {-a_{1} a_{3}}\, a_{0} +a_{1} a_{2}}{2 a_{1}}, a_{2} +1, 2 x \sqrt {-a_{1} a_{3}}\right ) c_1 \left (a_{2} \sqrt {-a_{1} a_{3}}+a_{3} a_{0} \right )\right )}
\]
Maple trace
Methods for first order ODEs:
--- Trying classification methods ---
trying a quadrature
trying 1st order linear
trying Bernoulli
trying separable
trying inverse linear
trying homogeneous types:
trying Chini
differential order: 1; looking for linear symmetries
trying exact
Looking for potential symmetries
trying Riccati
trying Riccati sub-methods:
<- Abel AIR successful: ODE belongs to the 1F1 2-parameter class
Maple step by step
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x \left (\frac {d}{d x}y \left (x \right )\right )+a_{3} x y \left (x \right )^{2}+a_{2} y \left (x \right )+a_{1} x +a_{0} =0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=-\frac {a_{3} x y \left (x \right )^{2}+a_{2} y \left (x \right )+a_{1} x +a_{0}}{x} \end {array} \]
2.2.35.3 ✓ Mathematica. Time used: 0.261 (sec). Leaf size: 421
ode=x*D[y[x],x]+a3*x*y[x]^2+a2*y[x]+a1*x+a0==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to -\frac {i \left (\sqrt {\text {a1}} c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\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 ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\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 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\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 ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}+2\right ),\text {a2}+1,2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i \sqrt {\text {a3}} \text {a0}}{\sqrt {\text {a1}}}+\text {a2}\right ),\text {a2},2 i \sqrt {\text {a1}} \sqrt {\text {a3}} x\right )}-i \sqrt {\text {a1}}}{\sqrt {\text {a3}}} \end{align*}
2.2.35.4 ✗ Sympy
from sympy import *
x = symbols("x")
a__0 = symbols("a__0")
a__1 = symbols("a__1")
a__2 = symbols("a__2")
a__3 = symbols("a__3")
y = Function("y")
ode = Eq(a__0 + a__1*x + a__2*y(x) + a__3*x*y(x)**2 + x*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
Timed Out