2.29.49 Problem 109

Internal problem ID [13770]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-3
Problem number : 109
Date solved : Sunday, January 18, 2026 at 09:14:34 PM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

2.29.49.1 second order linear exact ode

0.875 (sec)

\begin{align*} \left (x +\gamma \right ) y^{\prime \prime }+\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y&=0 \\ \end{align*}

Entering second order linear exact ode solverAn ode of the form

\begin{align*} p \left (x \right ) y^{\prime \prime }+q \left (x \right ) y^{\prime }+r \left (x \right ) y&=s \left (x \right ) \end{align*}

is exact if

\begin{align*} p''(x) - q'(x) + r(x) &= 0 \tag {1} \end{align*}

For the given ode the above values are

\begin{align*} p(x) &= x +\gamma \\ q(x) &= x^{n} a +b \,x^{m}+c\\ r(x) &= \frac {a n \,x^{n}+b \,x^{m} m}{x}\\ s(x) &= 0 \end{align*}

Hence

\begin{align*} p''(x) &= 0\\ q'(x) &= \frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x} \end{align*}

Therefore (1) becomes

\begin{align*} 0- \left (\frac {a n \,x^{n}}{x}+\frac {b \,x^{m} m}{x}\right ) + \left (\frac {a n \,x^{n}+b \,x^{m} m}{x}\right )&=0 \end{align*}

This shows the ode is exact. Since the ode is exact, it can be written as

\begin{align*} \left (p \left (x \right ) y^{\prime }+\left (q \left (x \right )-p^{\prime }\left (x \right )\right ) y\right )' &= s(x) \end{align*}

Integrating the above gives

\begin{align*} p \left (x \right ) y^{\prime }+\left (q \left (x \right )-p^{\prime }\left (x \right )\right ) y&=\int {s \left (x \right )\, dx} \end{align*}

Substituting the values of \(p,q,r,s\) into the above results in

\begin{align*} \left (x +\gamma \right ) y^{\prime }+\left (x^{n} a +b \,x^{m}+c -1\right ) y&=c_1 \end{align*}

Entering first order ode linear solverIn canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }\\ p(x) &=\frac {c_1}{x +\gamma } \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\\ &= {\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {c_1}{x +\gamma }\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) &= \left ({\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) \left (\frac {c_1}{x +\gamma }\right ) \\ \mathrm {d} \left (y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) &= \left (\frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }\right )\, \mathrm {d} x \\ \end{align*}

Integrating gives

\begin{align*} y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}&= \int {\frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma } \,dx} \\ &=\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x + c_2 \end{align*}

Dividing throughout by the integrating factor \({\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\) gives the final solution

\[ y = {\mathrm e}^{\int \frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \left (\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) \]

Summary of solutions found

\begin{align*} y &= {\mathrm e}^{\int \frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \left (\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) \\ \end{align*}

2.29.49.2 second order integrable as is

0.173 (sec)

\begin{align*} \left (x +\gamma \right ) y^{\prime \prime }+\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }+\left (a n \,x^{n -1}+b m \,x^{m -1}\right ) y&=0 \\ \end{align*}

Entering second order integrable as is solverIntegrating both sides of the ODE w.r.t \(x\) gives

\begin{align*} \int \left (\left (x +\gamma \right ) y^{\prime \prime }+\left (x^{n} a +b \,x^{m}+c \right ) y^{\prime }+\frac {\left (a n \,x^{n}+b \,x^{m} m \right ) y}{x}\right )d x &= 0 \\ \frac {\left (x^{n} a x +x^{m} b x +c x -x \right ) y}{x}+\left (x +\gamma \right ) y^{\prime } = c_1 \end{align*}

Which is now solved for \(y\). Entering first order ode linear solverIn canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }\\ p(x) &=\frac {c_1}{x +\gamma } \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\\ &= {\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu p \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {c_1}{x +\gamma }\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) &= \left ({\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) \left (\frac {c_1}{x +\gamma }\right ) \\ \mathrm {d} \left (y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\right ) &= \left (\frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }\right )\, \mathrm {d} x \\ \end{align*}

Integrating gives

\begin{align*} y \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}&= \int {\frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma } \,dx} \\ &=\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x + c_2 \end{align*}

Dividing throughout by the integrating factor \({\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}\) gives the final solution

\[ y = {\mathrm e}^{\int \frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \left (\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) \]

Summary of solutions found

\begin{align*} y &= {\mathrm e}^{\int \frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x} \left (\int \frac {c_1 \,{\mathrm e}^{\int -\frac {-x^{n} a -b \,x^{m}-c +1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) \\ \end{align*}

2.29.49.3 ✓ Maple. Time used: 0.001 (sec). Leaf size: 63

ode:=(x+gamma)*diff(diff(y(x),x),x)+(a*x^n+b*x^m+c)*diff(y(x),x)+(a*n*x^(n-1)+b*m*x^(m-1))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \int \frac {{\mathrm e}^{\int \frac {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x}}{x +\gamma }d x +c_2 \right ) {\mathrm e}^{-\int \frac {a \,x^{n}+b \,x^{m}+c -1}{x +\gamma }d x} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
<- linear_1 successful
 

2.29.49.4 ✗ Mathematica

ode=(x+\[Gamma])*D[y[x],{x,2}]+(a*x^n+b*x^m+c)*D[y[x],x]+(a*n*x^(n-1)+b*m*x^(m-1))*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.29.49.5 ✗ Sympy

from sympy import * 
x = symbols("x") 
Gamma = symbols("Gamma") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq((Gamma + x)*Derivative(y(x), (x, 2)) + (a*n*x**(n - 1) + b*m*x**(m - 1))*y(x) + (a*x**n + b*x**m + c)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

TypeError : Add object cannot be interpreted as an integer
 

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)) 
 
('factorable', '2nd_power_series_ordinary')