2.3.4 Problem 4

Internal problem ID [13284]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.3. Equations Containing Exponential Functions
Problem number : 4
Date solved : Sunday, January 18, 2026 at 07:07:49 PM
CAS classification : [_Riccati]

2.3.4.1 Solved using first_order_ode_riccati

0.334 (sec)

Entering first order ode riccati solver

\begin{align*} y^{\prime }&=\sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \\ \end{align*}

In canonical form the ODE is

\begin{align*} y' &= F(x,y)\\ &= \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \end{align*}

This is a Riccati ODE. Comparing the ODE to solve

\[ y' = \sigma y^{2}+a y+b \,{\mathrm e}^{x}+c \]

With Riccati ODE standard form

\[ y' = f_0(x)+ f_1(x)y+f_2(x)y^{2} \]

Shows that \(f_0(x)=b \,{\mathrm e}^{x}+c\), \(f_1(x)=a\) and \(f_2(x)=\sigma \). Let

\begin{align*} y &= \frac {-u'}{f_2 u} \\ &= \frac {-u'}{u \sigma } \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 &=a \sigma \\ f_2^2 f_0 &=\sigma ^{2} \left (b \,{\mathrm e}^{x}+c \right ) \end{align*}

Substituting the above terms back in equation (2) gives

\[ \sigma u^{\prime \prime }\left (x \right )-a \sigma u^{\prime }\left (x \right )+\sigma ^{2} \left (b \,{\mathrm e}^{x}+c \right ) u \left (x \right ) = 0 \]

Entering second order bessel ode form A solverWriting the ode as

\begin{align*} \sigma \left (\frac {d^{2}u}{d x^{2}}\right )-a \sigma \left (\frac {d u}{d x}\right )+\left ({\mathrm e}^{x} b \,\sigma ^{2}+c \,\sigma ^{2}\right ) u = 0\tag {1} \end{align*}

An ode of the form

\begin{equation} ay^{\prime \prime }+by^{\prime }+(ce^{rx}+m)y=0\tag {1}\end{equation}

can be transformed to Bessel ode using the transformation

\begin{align*} x & =\ln \left ( t\right ) \\ e^{x} & =t \end{align*}

Where \(a,b,c,m\) are not functions of \(x\) and where \(b\) and \(m\) are allowed to be be zero. Using this transformation gives

\begin{align} \frac {dy}{dx} & =\frac {dy}{dt}\frac {dt}{dx}\nonumber \\ & =\frac {dy}{dt}e^{x}\nonumber \\ & =t\frac {dy}{dt}\tag {2}\end{align}

And

\begin{align} \frac {d^{2}y}{dx^{2}} & =\frac {d}{dx}\left ( \frac {dy}{dx}\right ) \nonumber \\ & =\frac {d}{dx}\left ( t\frac {dy}{dt}\right ) \nonumber \\ & =\frac {d}{dt}\frac {dt}{dx}\left ( t\frac {dy}{dt}\right ) \nonumber \\ & =\frac {dt}{dx}\frac {d}{dt}\left ( t\frac {dy}{dt}\right ) \nonumber \\ & =t\frac {d}{dt}\left ( t\frac {dy}{dt}\right ) \nonumber \\ & =t\left ( \frac {dy}{dt}+t\frac {d^{2}y}{dt^{2}}\right ) \tag {3}\end{align}

Substituting (2,3) into (1) gives

\begin{align} at\left ( \frac {dy}{dt}+t\frac {d^{2}y}{dt^{2}}\right ) +bt\frac {dy}{dt}+(ce^{rx}+m)y & =0\nonumber \\ \left ( aty^{\prime }+at^{2}y^{\prime \prime }\right ) +bty^{\prime }+(ct^{r}+m)y & =0\nonumber \\ at^{2}y^{\prime \prime }+\left ( b+a\right ) ty^{\prime }+(ct^{r}+m)y & =0\nonumber \\ t^{2}y^{\prime \prime }+\frac {b+a}{a}ty^{\prime }+\left ( \frac {c}{a}t^{r}+\frac {m}{a}\right ) y & =0\tag {4}\end{align}

Which is Bessel ODE. Comparing the above to the general known Bowman form of Bessel ode which is

\begin{equation} t^{2}y^{\prime \prime }+\left ( 1-2\alpha \right ) ty^{\prime }+\left ( \beta ^{2}\gamma ^{2}t^{2\gamma }-\left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) \right ) y=0\tag {C}\end{equation}

And now comparing (4) and (C) shows that

\begin{align} \left ( 1-2\alpha \right ) & =\frac {b+a}{a}\tag {5}\\ \beta ^{2}\gamma ^{2} & =\frac {c}{a}\tag {6}\\ 2\gamma & =r\tag {7}\\ \left ( n^{2}\gamma ^{2}-\alpha ^{2}\right ) & =-\frac {m}{a}\tag {8}\end{align}

(5) gives \(\alpha =\frac {1}{2}-\frac {b+a}{2a}\). (7) gives \(\gamma =\frac {r}{2}\). (8) now becomes \(\left ( n^{2}\left ( \frac {r}{2}\right ) ^{2}-\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}\right ) =-\frac {m}{a}\) or \(n^{2}=\frac {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}{\left ( \frac {r}{2}\right ) ^{2}}\). Hence \(n=\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}\ \)by taking the positive root. And finally (6) gives \(\beta ^{2}=\frac {c}{a\gamma ^{2}}\) or \(\beta =\sqrt {\frac {c}{a}}\frac {1}{\gamma }=\sqrt {\frac {c}{a}}\frac {2}{r}\) (also taking the positive root). Hence

\begin{align*} \alpha & =\frac {1}{2}-\frac {b+a}{2a}\\ n & =\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}\\ \beta & =\sqrt {\frac {c}{a}}\frac {2}{r}\\ \gamma & =\frac {r}{2}\end{align*}

But the solution to (C) which is general form of Bessel ode is known and given by

\[ y\left ( t\right ) =t^{\alpha }\left ( c_{1}J_{n}\left ( \beta t^{\gamma }\right ) +c_{2}Y_{n}\left ( \beta t^{\gamma }\right ) \right ) \]

Substituting the above values found into this solution gives

\[ y\left ( t\right ) =t^{\frac {1}{2}-\frac {b+a}{2a}}\left ( c_{1}J_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}t^{\frac {r}{2}}\right ) +c_{2}Y_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}t^{\frac {r}{2}}\right ) \right ) \]

Since \(e^{x}=t\) then the above becomes

\begin{align} y\left ( x\right ) & =e^{x\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) }\left ( c_{1}J_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) +c_{2}Y_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {1}{2}-\frac {b+a}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) \right ) \nonumber \\ & =e^{x\left ( \frac {-b}{2a}\right ) }\left ( c_{1}J_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {-b}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) +c_{2}Y_{\frac {2}{r}\sqrt {-\frac {m}{a}+\left ( \frac {-b}{2a}\right ) ^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) \right ) \nonumber \\ & =e^{x\left ( \frac {-b}{2a}\right ) }\left ( c_{1}J_{\frac {2}{r}\sqrt {-\frac {m}{a}+\frac {b^{2}}{4a^{2}}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) +c_{2}Y_{\frac {2}{r}\sqrt {-\frac {m}{a}+\frac {b^{2}}{4a^{2}}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) \right ) \nonumber \\ & =e^{x\left ( \frac {-b}{2a}\right ) }\left ( c_{1}J_{\frac {2}{r}\sqrt {-\frac {4ma+b^{2}}{4a^{2}}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) +c_{2}Y_{\frac {2}{r}\sqrt {-\frac {4ma+b^{2}}{4a^{2}}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) \right ) \nonumber \\ & =e^{x\left ( \frac {-b}{2a}\right ) }\left ( c_{1}J_{\frac {1}{ra}\sqrt {-4ma+b^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) +c_{2}Y_{\frac {1}{ra}\sqrt {-4ma+b^{2}}}\left ( \sqrt {\frac {c}{a}}\frac {2}{r}e^{x\frac {r}{2}}\right ) \right ) \tag {9}\end{align}

Equation (9) above is the solution to \(ay^{\prime \prime }+by^{\prime }+(ce^{rx}+m)y=0\). Therefore we just need now to compare this form to the ode given and use (9) to obtain the final solution.

Comparing form (1) to the ode we are solving shows that

\begin{align*} a &= \sigma \\ b &= -a \sigma \\ c &= \sigma ^{2} b\\ r &= 1\\ m &= c \,\sigma ^{2} \end{align*}

Substituting these in (9) gives the solution as

\begin{align*} u&=c_1 \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+c_2 \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) \end{align*}

Taking derivative gives

\begin{equation} \tag{4} u^{\prime }\left (x \right ) = \frac {c_1 a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+c_1 \,{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}+\frac {c_2 a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+c_2 \,{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}} \end{equation}

Substituting equations (3,4) into (1) results in

\begin{align*} y &= \frac {-u'}{f_2 u} \\ y &= \frac {-u'}{u \sigma } \\ y &= -\frac {\frac {c_1 a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+c_1 \,{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}+\frac {c_2 a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+c_2 \,{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}}{\sigma \left (c_1 \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+c_2 \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \\ \end{align*}

Doing change of constants, the above solution becomes

\[ y = -\frac {\frac {a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}+\frac {c_3 a \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2}+c_3 \,{\mathrm e}^{\frac {a x}{2}} \left (-\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+\frac {\sqrt {a^{2}-4 c \sigma }\, {\mathrm e}^{-\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )}{2 \sqrt {\sigma b}}\right ) \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}}{\sigma \left ({\mathrm e}^{\frac {a x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )+c_3 \,{\mathrm e}^{\frac {a x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \]

Simplifying the above gives

\begin{align*} y &= \frac {2 \sqrt {\sigma b}\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) {\mathrm e}^{\frac {x}{2}}-\left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 c \sigma }+a \right )}{2 \sigma \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \\ \end{align*}

Summary of solutions found

\begin{align*} y &= \frac {2 \sqrt {\sigma b}\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }+1, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) {\mathrm e}^{\frac {x}{2}}-\left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 c \sigma }+a \right )}{2 \sigma \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_3 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 c \sigma }, 2 \sqrt {\sigma b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \\ \end{align*}

2.3.4.2 ✓ Maple. Time used: 0.003 (sec). Leaf size: 200

ode:=diff(y(x),x) = sigma*y(x)^2+a*y(x)+b*exp(x)+c; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {-2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}\, \sigma -2 \,{\mathrm e}^{\frac {x}{2}} \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}+1, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) \sqrt {b}\, c_1 \sigma +\sqrt {\sigma }\, \left (\operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_1 +\operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right ) \left (\sqrt {a^{2}-4 \sigma c}+a \right )}{\sigma ^{{3}/{2}} \left (2 \operatorname {BesselY}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right ) c_1 +2 \operatorname {BesselJ}\left (\sqrt {a^{2}-4 \sigma c}, 2 \sqrt {\sigma }\, \sqrt {b}\, {\mathrm e}^{\frac {x}{2}}\right )\right )} \]

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: 
   trying Riccati_symmetries 
   trying Riccati to 2nd Order 
   -> Calling odsolve with the ODE, diff(diff(y(x),x),x) = a*diff(y(x),x)-sigma 
*(b*exp(x)+c)*y(x), y(x) 
      *** Sublevel 2 *** 
      Methods for second order ODEs: 
      --- Trying classification methods --- 
      trying a symmetry of the form [xi=0, eta=F(x)] 
      checking if the LODE is missing y 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
      -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int\ 
(r(x), dx)) * 2F1([a1, a2], [b1], f) 
      -> Trying changes of variables to rationalize or make the ODE simpler 
         trying a quadrature 
         checking if the LODE has constant coefficients 
         checking if the LODE is of Euler type 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         checking if the LODE is missing y 
         -> Trying a Liouvillian solution using Kovacics algorithm 
         <- No Liouvillian solutions exists 
         -> Trying a solution in terms of special functions: 
            -> Bessel 
            <- Bessel successful 
         <- special function solution successful 
         Change of variables used: 
            [x = ln(t)] 
         Linear ODE actually solved: 
            (b*sigma*t+c*sigma)*u(t)+(-a*t+t)*diff(u(t),t)+t^2*diff(diff(u(t),t\ 
),t) = 0 
      <- change of variables successful 
   <- Riccati to 2nd Order successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\sigma y \left (x \right )^{2}+a y \left (x \right )+b \,{\mathrm e}^{x}+c \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\sigma y \left (x \right )^{2}+a y \left (x \right )+b \,{\mathrm e}^{x}+c \end {array} \]

2.3.4.3 ✓ Mathematica. Time used: 0.337 (sec). Leaf size: 546

ode=D[y[x],x]==sigma*y[x]^2+a*y[x]+b*Exp[x]+c; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {a \sqrt {b \sigma e^x} \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+b \sigma e^x \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )-b \sigma e^x \operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma }+1,2 \sqrt {b e^x \sigma }\right )+a c_1 \sqrt {b \sigma e^x} \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+b c_1 \sigma e^x \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )-b c_1 \sigma e^x \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (1-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}{2 \sigma \sqrt {b \sigma e^x} \left (\operatorname {Gamma}\left (\sqrt {a^2-4 c \sigma }+1\right ) \operatorname {BesselJ}\left (\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )+c_1 \operatorname {Gamma}\left (1-\sqrt {a^2-4 c \sigma }\right ) \operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )\right )}\\ y(x)&\to \frac {\frac {\sqrt {b \sigma e^x} \left (\operatorname {BesselJ}\left (1-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )-\operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma }-1,2 \sqrt {b e^x \sigma }\right )\right )}{\operatorname {BesselJ}\left (-\sqrt {a^2-4 c \sigma },2 \sqrt {b e^x \sigma }\right )}-a}{2 \sigma } \end{align*}

2.3.4.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
sigma = symbols("sigma") 
y = Function("y") 
ode = Eq(-a*y(x) - b*exp(x) - c - sigma*y(x)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a*y(x) - b*exp(x) - c - sigma*y(x)**2 + Derivative(y(x), x) can
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('1st_power_series', 'lie_group')