2.8.14 Problem 23

Internal problem ID [13366]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.2. Riccati Equation. subsection 1.2.5-2
Problem number : 23
Date solved : Wednesday, December 31, 2025 at 01:58:41 PM
CAS classification : [_Riccati]

2.8.14.1 Solved using first_order_ode_riccati_by_guessing_particular_solution

0.480 (sec)

Entering first order ode riccati guess solver

\begin{align*} \left (a \ln \left (x \right )+b \right ) y^{\prime }&=\ln \left (x \right )^{n} y^{2}+c y-\lambda ^{2} \ln \left (x \right )^{n}+c \lambda \\ \end{align*}

This is a Riccati ODE. Comparing the above ODE to solve with the Riccati standard form

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

Shows that

\begin{align*} f_0(x) & =-\frac {\lambda ^{2} \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}+\frac {c \lambda }{a \ln \left (x \right )+b}\\ f_1(x) & =\frac {c}{a \ln \left (x \right )+b}\\ f_2(x) &=\frac {\ln \left (x \right )^{n}}{a \ln \left (x \right )+b} \end{align*}

Using trial and error, the following particular solution was found

\[ y_p = -\lambda \]

Since a particular solution is known, then the general solution is given by

\begin{align*} y &= y_p + \frac { \phi (x) }{ c_1 - \int { \phi (x) f_2 \,dx} } \end{align*}

Where

\begin{align*} \phi (x) &= e^{ \int 2 f_2 y_p + f_1 \,dx } \end{align*}

Evaluating the above gives the general solution as

\[ y = -\lambda +\frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}+\frac {c}{a \ln \left (x \right )+b}\right )d x}}{c_1 -\int \frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}+\frac {c}{a \ln \left (x \right )+b}\right )d x} \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}d x} \]

Summary of solutions found

\begin{align*} y &= -\lambda +\frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}+\frac {c}{a \ln \left (x \right )+b}\right )d x}}{c_1 -\int \frac {{\mathrm e}^{\int \left (-\frac {2 \lambda \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}+\frac {c}{a \ln \left (x \right )+b}\right )d x} \ln \left (x \right )^{n}}{a \ln \left (x \right )+b}d x} \\ \end{align*}

2.8.14.2 ✓ Maple. Time used: 0.006 (sec). Leaf size: 124

ode:=(a*ln(x)+b)*diff(y(x),x) = ln(x)^n*y(x)^2+c*y(x)-lambda^2*ln(x)^n+lambda*c; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-\lambda \int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x -c_1 \lambda -{\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{c_1 +\int \frac {\ln \left (x \right )^{n} {\mathrm e}^{-\int \frac {2 \ln \left (x \right )^{n} \lambda -c}{a \ln \left (x \right )+b}d x}}{a \ln \left (x \right )+b}d x} \]

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) = (ln(x)*a*n+c*x*ln(x) 
-a*ln(x)+b*n)/x/ln(x)/(a*ln(x)+b)*diff(y(x),x)+ln(x)^n*lambda*(ln(x)^n*lambda-c 
)/(a*ln(x)+b)^2*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 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 a symmetry of the form [xi=0, eta=F(x)] 
            trying 2nd order exact linear 
            trying symmetries linear in x and y(x) 
            trying to convert to a linear ODE with constant coefficients 
      <- unable to find a useful change of variables 
         trying a symmetry of the form [xi=0, eta=F(x)] 
         trying 2nd order exact linear 
         trying symmetries linear in x and y(x) 
         trying to convert to a linear ODE with constant coefficients 
         trying 2nd order, integrating factor of the form mu(x,y) 
         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 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 und\ 
er a power @ Moebius 
            -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = e\ 
xp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) 
               trying a symmetry of the form [xi=0, eta=F(x)] 
               trying 2nd order exact linear 
               trying symmetries linear in x and y(x) 
               trying to convert to a linear ODE with constant coefficients 
         <- unable to find a useful change of variables 
            trying a symmetry of the form [xi=0, eta=F(x)] 
         trying to convert to an ODE of Bessel type 
   -> Trying a change of variables to reduce to Bernoulli 
   -> Calling odsolve with the ODE, diff(y(x),x)-(1/(a*ln(x)+b)*ln(x)^n*y(x)^2+ 
y(x)+1/(a*ln(x)+b)*c*y(x)*x+x^2*(-1/(a*ln(x)+b)*lambda^2*ln(x)^n+1/(a*ln(x)+b)* 
c*lambda))/x, y(x), explicit 
      *** Sublevel 2 *** 
      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 inverse_Riccati 
      trying 1st order ODE linearizable_by_differentiation 
   -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
   -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
   <- symmetry pattern of the form [0, F(x)*G(y)] successful 
   <- Riccati with symmetry pattern of the form [0,F(x)*G(y)] successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a \ln \left (x \right )+b \right ) \left (\frac {d}{d x}y \left (x \right )\right )=\ln \left (x \right )^{13366} y \left (x \right )^{2}+c y \left (x \right )-\lambda ^{2} \ln \left (x \right )^{13366}+c \lambda \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\ln \left (x \right )^{13366} y \left (x \right )^{2}+c y \left (x \right )-\lambda ^{2} \ln \left (x \right )^{13366}+c \lambda }{a \ln \left (x \right )+b} \end {array} \]

2.8.14.3 ✓ Mathematica. Time used: 1.346 (sec). Leaf size: 286

ode=(a*Log[x]+b)*D[y[x],x]==(Log[x])^n*y[x]^2+c*y[x]-\[Lambda]^2*(Log[x])^n+c*\[Lambda]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^x\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (-\lambda \log ^n(K[2])+y(x) \log ^n(K[2])+c\right )}{c n (b+a \log (K[2])) (\lambda +y(x))}dK[2]+\int _1^{y(x)}\left (-\int _1^x\left (\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \log ^n(K[2])}{c n (\lambda +K[3]) (b+a \log (K[2]))}-\frac {\exp \left (-\int _1^{K[2]}-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right ) \left (-\lambda \log ^n(K[2])+K[3] \log ^n(K[2])+c\right )}{c n (\lambda +K[3])^2 (b+a \log (K[2]))}\right )dK[2]-\frac {\exp \left (-\int _1^x-\frac {c-2 \lambda \log ^n(K[1])}{b+a \log (K[1])}dK[1]\right )}{c n (\lambda +K[3])^2}\right )dK[3]=c_1,y(x)\right ] \]

2.8.14.4 ✗ Sympy

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

Timed Out