2.32.28 Problem 210

Internal problem ID [13870]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 210
Date solved : Thursday, January 01, 2026 at 03:14:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

2.32.28.1 second order change of variable on x method 2

2.450 (sec)

\begin{align*} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (-k +1\right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} y&=0 \\ \end{align*}

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

\begin{align*} 2 x \left (a \,x^{2}+b x +c \right ) y^{\prime \prime }+\left (a \left (2-k \right ) x^{2}+b \left (-k +1\right ) x -c k \right ) y^{\prime }+\lambda \,x^{k +1} 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 {a \left (k -2\right ) x^{2}+b \left (k -1\right ) x +c k}{2 x \left (a \,x^{2}+b x +c \right )}\\ q \left (x \right )&=\frac {\lambda \,x^{k +1}}{2 a \,x^{3}+2 b \,x^{2}+2 c x} \end{align*}

Applying change of variables \(\tau = g \left (x \right )\) to (2) gives

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end{align*}

Where \(\tau \) is the new independent variable, and

\begin{align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end{align*}

Let \(p_{1} = 0\). Eq (4) simplifies to

\begin{align*} \tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )&=0 \end{align*}

This ode is solved resulting in

\begin{align*} \tau &= \int {\mathrm e}^{-\int p \left (x \right )d x}d x\\ &= \int {\mathrm e}^{-\int -\frac {a \left (k -2\right ) x^{2}+b \left (k -1\right ) x +c k}{2 x \left (a \,x^{2}+b x +c \right )}d x}d x\\ &= \int e^{-\frac {\ln \left (a \,x^{2}+b x +c \right )}{2}+\frac {\ln \left (x \right ) k}{2}} \,dx\\ &= \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x\\ &= \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x\tag {6} \end{align*}

Using (6) to evaluate \(q_{1}\) from (5) gives

\begin{align*} q_{1} \left (\tau \right ) &= \frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &= \frac {\frac {\lambda \,x^{k +1}}{2 a \,x^{3}+2 b \,x^{2}+2 c x}}{\frac {x^{k}}{a \,x^{2}+b x +c}}\\ &= \frac {\lambda }{2}\tag {7} \end{align*}

Substituting the above in (3) and noting that now \(p_{1} = 0\) results in

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+q_{1} y \left (\tau \right )&=0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+\frac {\lambda y \left (\tau \right )}{2}&=0 \end{align*}

The above ode is now solved for \(y \left (\tau \right )\).Entering second order linear constant coefficient ode solver

This is second order with constant coefficients homogeneous ODE. In standard form the ODE is

\[ A y''(\tau ) + B y'(\tau ) + C y(\tau ) = 0 \]

Where in the above \(A=1, B=0, C=\frac {\lambda }{2}\). Let the solution be \(y \left (\tau \right )=e^{\lambda \tau }\). Substituting this into the ODE gives

\[ \lambda ^{2} {\mathrm e}^{\tau \lambda }+\frac {\lambda \,{\mathrm e}^{\tau \lambda }}{2} = 0 \tag {1} \]

Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda \tau }\) gives

\[ \lambda ^{2}+\frac {\lambda }{2} = 0 \tag {2} \]

Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula

\[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \]

Substituting \(A=1, B=0, C=\frac {\lambda }{2}\) into the above gives

\begin{align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {0^2 - (4) \left (1\right )\left (\frac {\lambda }{2}\right )}\\ &= \pm \frac {\sqrt {-2 \lambda }}{2} \end{align*}

Hence

\begin{align*} \lambda _1 &= + \frac {\sqrt {-2 \lambda }}{2} \\ \lambda _2 &= - \frac {\sqrt {-2 \lambda }}{2} \\ \end{align*}

Which simplifies to

\begin{align*} \lambda _1 &= \frac {\sqrt {2}\, \sqrt {-\lambda }}{2} \\ \lambda _2 &= -\frac {\sqrt {2}\, \sqrt {-\lambda }}{2} \\ \end{align*}

Since roots are distinct, then the solution is

\begin{align*} y \left (\tau \right ) &= c_1 e^{\lambda _1 \tau } + c_2 e^{\lambda _2 \tau } \\ y \left (\tau \right ) &= c_1 e^{\left (\frac {\sqrt {2}\, \sqrt {-\lambda }}{2}\right )\tau } +c_2 e^{\left (-\frac {\sqrt {2}\, \sqrt {-\lambda }}{2}\right )\tau } \\ \end{align*}

Or

\[ y \left (\tau \right ) =c_1 \,{\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\lambda }\, \tau }{2}}+c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\lambda }\, \tau }{2}} \]

The above solution is now transformed back to \(y\) using (6) which results in

\[ y = c_1 \,{\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\lambda }\, \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x}{2}}+c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\lambda }\, \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x}{2}} \]

Summary of solutions found

\begin{align*} y &= c_1 \,{\mathrm e}^{\frac {\sqrt {2}\, \sqrt {-\lambda }\, \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x}{2}}+c_2 \,{\mathrm e}^{-\frac {\sqrt {2}\, \sqrt {-\lambda }\, \int \frac {x^{\frac {k}{2}}}{\sqrt {a \,x^{2}+b x +c}}d x}{2}} \\ \end{align*}

2.32.28.2 ✓ Maple. Time used: 0.001 (sec). Leaf size: 65

ode:=2*x*(a*x^2+b*x+c)*diff(diff(y(x),x),x)+(a*(2-k)*x^2+b*(1-k)*x-c*k)*diff(y(x),x)+lambda*x^(1+k)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_1 \,{\mathrm e}^{i \sqrt {2}\, \int \sqrt {\frac {\lambda \,x^{k}}{a \,x^{2}+b x +c}}d x}+c_2 \right ) {\mathrm e}^{-\frac {i \sqrt {2}\, \int \sqrt {\frac {\lambda \,x^{k}}{a \,x^{2}+b x +c}}d x}{2}} \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a symmetry of the form [xi=0, eta=F(x)] 
   Solution is available but has integrals. Trying a simpler solution using Kov\ 
acics algorithm... 
   Solution via Kovacic is not simpler. Returning default solution 
   <- linear_1 successful
 

2.32.28.3 ✓ Mathematica. Time used: 107.117 (sec). Leaf size: 774

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

\begin{align*} y(x)&\to \frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}{\sqrt {-\sec ^2\left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}-c_2\right )}}\\ y(x)&\to -\frac {\sqrt {2} \sqrt {c_1} \tan \left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}{\sqrt {-\sec ^2\left (\frac {x \sqrt {\frac {\sqrt {b^2-4 a c}+2 a x+b}{\sqrt {b^2-4 a c}+b}} \sqrt {\frac {-2 \sqrt {b^2-4 a c}+4 a x+2 b}{b-\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {k+2}{2},\frac {1}{2},\frac {1}{2},\frac {k+4}{2},-\frac {2 a x}{b+\sqrt {b^2-4 a c}},\frac {2 a x}{\sqrt {b^2-4 a c}-b}\right )}{(k+2) \sqrt {\frac {x^{-k} (x (a x+b)+c)}{\lambda }}}+c_2\right )}} \end{align*}

2.32.28.4 ✗ Sympy

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

ValueError : Expected Expr or iterable but got None