2.29.43 Problem 103

Internal problem ID [13764]
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 : 103
Date solved : Thursday, January 01, 2026 at 02:33:27 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

2.29.43.1 second order change of variable on y method 2

1.619 (sec)

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

Entering second order change of variable on \(y\) method 2 solverIn normal form the ode

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

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=\frac {x^{n} a +b \,x^{m}+c}{x}\\ q \left (x \right )&=\frac {\left (c -1\right ) \left (x^{n} a +b \,x^{m}\right )}{x^{2}} \end{align*}

Applying change of variables on the depndent variable \(y = v \left (x \right ) x^{n}\) to (2) gives the following ode where the dependent variables is \(v \left (x \right )\) and not \(y\).

\begin{align*} v^{\prime \prime }\left (x \right )+\left (\frac {2 n}{x}+p \right ) v^{\prime }\left (x \right )+\left (\frac {n \left (n -1\right )}{x^{2}}+\frac {n p}{x}+q \right ) v \left (x \right )&=0 \tag {3} \end{align*}

Let the coefficient of \(v \left (x \right )\) above be zero. Hence

\begin{align*} \frac {n \left (n -1\right )}{x^{2}}+\frac {n p}{x}+q&=0 \tag {4} \end{align*}

Substituting the earlier values found for \(p \left (x \right )\) and \(q \left (x \right )\) into (4) gives

\begin{align*} \frac {n \left (n -1\right )}{x^{2}}+\frac {n \left (x^{n} a +b \,x^{m}+c \right )}{x^{2}}+\frac {\left (c -1\right ) \left (x^{n} a +b \,x^{m}\right )}{x^{2}}&=0 \tag {5} \end{align*}

Solving (5) for \(n\) gives

\begin{align*} n&=-c +1 \tag {6} \end{align*}

Substituting this value in (3) gives

\begin{align*} v^{\prime \prime }\left (x \right )+\left (\frac {-2 c +2}{x}+\frac {x^{n} a +b \,x^{m}+c}{x}\right ) v^{\prime }\left (x \right )&=0 \\ v^{\prime \prime }\left (x \right )+\frac {\left (-c +2+x^{n} a +b \,x^{m}\right ) v^{\prime }\left (x \right )}{x}&=0 \tag {7} \\ \end{align*}

Using the substitution

\begin{align*} u \left (x \right ) = v^{\prime }\left (x \right ) \end{align*}

Then (7) becomes

\begin{align*} u^{\prime }\left (x \right )+\frac {\left (-c +2+x^{n} a +b \,x^{m}\right ) u \left (x \right )}{x} = 0 \tag {8} \\ \end{align*}

The above is now solved for \(u \left (x \right )\). Entering first order ode linear solverIn canonical form a linear first order is

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

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\frac {c -2-x^{n} a -b \,x^{m}}{x}\\ p(x) &=0 \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {c -2-x^{n} a -b \,x^{m}}{x}d x}\\ &= x^{2-c} {\mathrm e}^{\frac {a m \,x^{n}+x^{m} b n}{m n}} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu u &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (u \,x^{2-c} {\mathrm e}^{\frac {a m \,x^{n}+x^{m} b n}{m n}}\right ) &= 0 \end{align*}

Integrating gives

\begin{align*} u \,x^{2-c} {\mathrm e}^{\frac {a m \,x^{n}+x^{m} b n}{m n}}&= \int {0 \,dx} + c_1 \\ &=c_1 \end{align*}

Dividing throughout by the integrating factor \(x^{2-c} {\mathrm e}^{\frac {a m \,x^{n}+x^{m} b n}{m n}}\) gives the final solution

\[ u \left (x \right ) = x^{-2+c} {\mathrm e}^{-\frac {a m \,x^{n}+x^{m} b n}{m n}} c_1 \]

Simplifying the above gives

\begin{align*} u \left (x \right ) &= x^{-2+c} {\mathrm e}^{\frac {-a m \,x^{n}-x^{m} b n}{n m}} c_1 \\ \end{align*}

Now that \(u \left (x \right )\) is known, then

\begin{align*} v^{\prime }\left (x \right )&= u \left (x \right )\\ v \left (x \right )&= \int u \left (x \right )d x +c_2\\ &= \int x^{-2+c} {\mathrm e}^{\frac {-a m \,x^{n}-x^{m} b n}{n m}} c_1 d x +c_2 \end{align*}

Hence

\begin{align*} y&= v \left (x \right ) x^{n}\\ &= \left (\int x^{-2+c} {\mathrm e}^{\frac {-a m \,x^{n}-x^{m} b n}{n m}} c_1 d x +c_2 \right ) x^{-c +1}\\ &= x^{-c +1} \left (c_1 \int x^{-2+c} {\mathrm e}^{-\frac {b \,x^{m}}{m}-\frac {a \,x^{n}}{n}}d x +c_2 \right )\\ \end{align*}

Summary of solutions found

\begin{align*} y &= \left (\int x^{-2+c} {\mathrm e}^{\frac {-a m \,x^{n}-x^{m} b n}{n m}} c_1 d x +c_2 \right ) x^{-c +1} \\ \end{align*}

2.29.43.2 ✗ Maple

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

\[ \text {No solution found} \]

Maple trace

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 
-> Trying an equivalence, under non-integer power transformations, 
   to LODEs admitting Liouvillian solutions. 
-> Trying a solution in terms of special functions: 
   -> Bessel 
   -> elliptic 
   -> Legendre 
   -> Kummer 
      -> hyper3: Equivalence to 1F1 under a power @ Moebius 
   -> hypergeometric 
      -> heuristic approach 
      -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
   -> Mathieu 
      -> Equivalence to the rational form of Mathieu ODE under a power @ Moebi\ 
us 
-> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power \ 
@ Moebius 
-> 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 
<- unable to find a useful change of variables 
   trying a symmetry of the form [xi=0, eta=F(x)] 
   trying differential order: 2; exact nonlinear 
   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 
   -> Trying an equivalence, under non-integer power transformations, 
      to LODEs admitting Liouvillian solutions. 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Whittaker 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Mo\ 
ebius 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ 
er @ Moebius 
   -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a pow\ 
er @ 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 
   <- 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 reduction of order to Riccati 
      trying Riccati sub-methods: 
         trying Riccati_symmetries 
         -> trying a symmetry pattern of the form [F(x)*G(y), 0] 
         -> trying a symmetry pattern of the form [0, F(x)*G(y)] 
         -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] 
--- Trying Lie symmetry methods, 2nd order --- 
   -> Computing symmetries using: way = 3 
[0, y] 
   <- successful computation of symmetries. 
   -> Computing symmetries using: way = 5
 

2.29.43.3 ✗ Mathematica

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

Not solved

2.29.43.4 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
m = symbols("m") 
n = symbols("n") 
y = Function("y") 
ode = Eq(x*Derivative(y(x), (x, 2)) + (c - 1)*(a*x**(n - 1) + b*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)
 

ValueError : Expected Expr or iterable but got None