2.1.298 Problem 301

Internal problem ID [9468]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 301
Date solved : Friday, April 25, 2025 at 06:09:05 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

$$\begin{array}{rl}x{(x-1)}^2{y}^{\prime \prime }-2y& =0\end{array}$$

Solved as second order ode using Kovacic algorithm

Time used: 0.222 (sec)

Writing the ode as

$$\begin{array}{}\text{(1)}& x{(x-1)}^2{y}^{\prime \prime }-2y& =0\text{(2)}& A{y}^{\prime \prime }+B{y}^{\prime }+Cy& =0\end{array}$$

Comparing (1) and (2) shows that

$$\begin{array}{rl}A& =x{(x-1)}^2\\ \text{(3)}& B& =0C& =-2\end{array}$$

Applying the Liouville transformation on the dependent variable gives

$$\begin{array}{rl}z(x)& =y{e}^{\int \frac{B}{2A}dx}\end{array}$$

Then (2) becomes

$$\begin{array}{}\text{(4)}& z^{″}(x)=rz(x)\end{array}$$

Where $r$ is given by

$$\begin{array}{rl}\text{(5)}& r& =\frac{s}{t}& =\frac{2A{B}^{\prime }-2B{A}^{\prime }+B^2-4AC}{4A^2}\end{array}$$

Substituting the values of $A,B,C$ from (3) in the above and simplifying gives

$$\begin{array}{}\text{(6)}& r& =\frac{2}{x{(x-1)}^2}\end{array}$$

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =2\\ t& =x{(x-1)}^2\end{array}$$

Therefore eq. (4) becomes

$$\begin{array}{}\text{(7)}& z^{″}(x)& =\left(\frac{2}{x{(x-1)}^2}\right)z(x)\end{array}$$

Equation (7) is now solved. After finding $z(x)$ then $y$ is found using the inverse transformation

$$\begin{array}{rl}y& =z\left(x\right)e^{-\int \frac{B}{2A}dx}\end{array}$$

The first step is to determine the case of Kovacic algorithm this ode belongs to. There are 3 cases depending on the order of poles of $r$ and the order of $r$ at $\mathrm{\infty }$. The following table summarizes these cases.

Case	Allowed pole order for $r$	Allowed value for $\mathcal{O}(\mathrm{\infty })$
1	$\{0,1,2,4,6,8,\cdots \}$	$\{\cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \}$
2	Need to have at least one pole that is either order $2$ or odd order greater than $2$. Any other pole order is allowed as long as the above condition is satisfied. Hence the following set of pole orders are all allowed. $\{1,2\}$,$\{1,3\}$,$\{2\}$,$\{3\}$,$\{3,4\}$,$\{1,2,5\}$.	no condition
3	$\{1,2\}$	$\{2,3,4,5,6,7,\cdots \}$

Table 2.298: Necessary conditions for each Kovacic case

The order of $r$ at $\mathrm{\infty }$ is the degree of $t$ minus the degree of $s$. Therefore

$$\begin{array}{rl}O\left(\mathrm{\infty }\right)& =\text{deg}(t)-\text{deg}(s)\\ & =3-0\\ & =3\end{array}$$

The poles of $r$ in eq. (7) and the order of each pole are determined by solving for the roots of $t=x{(x-1)}^2$. There is a pole at $x=0$ of order $1$. There is a pole at $x=1$ of order $2$. Since there is no odd order pole larger than $2$ and the order at $\mathrm{\infty }$ is $3$ then the necessary conditions for case one are met. Since there is a pole of order $2$ then necessary conditions for case two are met. Since pole order is not larger than $2$ and the order at $\mathrm{\infty }$ is $3$ then the necessary conditions for case three are met. Therefore

$$\begin{array}{rl}L& =[1,2,4,6,12]\end{array}$$

Attempting to find a solution using case $n=1$.

Looking at poles of order 1. For the pole at $x=0$ of order 1 then

$$\begin{array}{rl}[\sqrt{r}{]}_c& =0\\ {\alpha }_c^{+}& =1\\ {\alpha }_c^{-}& =1\end{array}$$

Looking at poles of order 2. The partial fractions decomposition of $r$ is

$$r=\frac{2}{{(x-1)}^2}+\frac{2}{x}-\frac{2}{x-1}$$

For the pole at $x=1$ let $b$ be the coefficient of $\frac{1}{{(x-1)}^2}$ in the partial fractions decomposition of $r$ given above. Therefore $b=2$. Hence

$$\begin{array}{rl}[\sqrt{r}{]}_c& =0\\ {\alpha }_c^{+}& =\frac{1}{2}+\sqrt{1+4b}& & =2\\ {\alpha }_c^{-}& =\frac{1}{2}-\sqrt{1+4b}& & =-1\end{array}$$

Since the order of $r$ at $\mathrm{\infty }$ is $3>2$ then

$$\begin{array}{rl}[\sqrt{r}{]}_{\mathrm{\infty }}& =0\\ {\alpha }_{\mathrm{\infty }}^{+}& =0\\ {\alpha }_{\mathrm{\infty }}^{-}& =1\end{array}$$

The following table summarizes the findings so far for poles and for the order of $r$ at $\mathrm{\infty }$ where $r$ is

$$r=\frac{2}{x{(x-1)}^2}$$

pole $c$ location	pole order	$[\sqrt{r}{]}_c$	${\alpha }_c^{+}$	${\alpha }_c^{-}$
$0$	$1$	$0$	$0$	$1$
$1$	$2$	$0$	$2$	$-1$

Order of $r$ at $\mathrm{\infty }$	$[\sqrt{r}{]}_{\mathrm{\infty }}$	${\alpha }_{\mathrm{\infty }}^{+}$	${\alpha }_{\mathrm{\infty }}^{-}$
$3$	$0$	$0$	$1$

Now that the all $[\sqrt{r}{]}_c$ and its associated ${\alpha }_c^{\pm }$ have been determined for all the poles in the set $\mathrm{\Gamma }$ and $[\sqrt{r}{]}_{\mathrm{\infty }}$ and its associated ${\alpha }_{\mathrm{\infty }}^{\pm }$ have also been found, the next step is to determine possible non negative integer $d$ from these using

$$\begin{array}{rl}d& ={\alpha }_{\mathrm{\infty }}^{s(\mathrm{\infty })}-\sum _{c\in \mathrm{\Gamma }}{\alpha }_c^{s(c)}\end{array}$$

Where $s(c)$ is either $+$ or $-$ and $s(\mathrm{\infty })$ is the sign of ${\alpha }_{\mathrm{\infty }}^{\pm }$. This is done by trial over all set of families $s=(s(c){)}_{c\in \mathrm{\Gamma }\cup \mathrm{\infty }}$ until such $d$ is found to work in finding candidate $\omega$. Trying ${\alpha }_{\mathrm{\infty }}^{+}=0$ then

$$\begin{array}{rl}d& ={\alpha }_{\mathrm{\infty }}^{+}-({\alpha }_{c_1}^{-}+{\alpha }_{c_2}^{-})\\ & =0-\left(0\right)\\ & =0\end{array}$$

Since $d$ an integer and $d\ge 0$ then it can be used to find $\omega$ using

$$\begin{array}{rl}\omega & =\sum _{c\in \mathrm{\Gamma }}(s(c)[\sqrt{r}{]}_c+\frac{{\alpha }_c^{s(c)}}{x-c})+s(\mathrm{\infty })[\sqrt{r}{]}_{\mathrm{\infty }}\end{array}$$

Substituting the above values in the above results in

$$\begin{array}{rl}\omega & =((-)[\sqrt{r}{]}_{c_1}+\frac{{\alpha }_{c_1}^{-}}{x-c_1})+((-)[\sqrt{r}{]}_{c_2}+\frac{{\alpha }_{c_2}^{-}}{x-c_2})+(+)[\sqrt{r}{]}_{\mathrm{\infty }}\\ & =\frac{1}{x}-\frac{1}{x-1}+\left(0\right)\\ & =\frac{1}{x}-\frac{1}{x-1}\\ & =-\frac{1}{x(x-1)}\end{array}$$

Now that $\omega$ is determined, the next step is find a corresponding minimal polynomial $p(x)$ of degree $d=0$ to solve the ode. The polynomial $p(x)$ needs to satisfy the equation

$$\begin{array}{}\text{(1A)}& p^{″}+2\omega p^{\prime }+({\omega }^{\prime }+{\omega }^2-r)p=0\end{array}$$

Let

$$\begin{array}{}\text{(2A)}& p(x)& =1\end{array}$$

Substituting the above in eq. (1A) gives

$$\begin{array}{rl}\left(0\right)+2(\frac{1}{x}-\frac{1}{x-1})\left(0\right)+((-\frac{1}{x^2}+\frac{1}{{(x-1)}^2})+{(\frac{1}{x}-\frac{1}{x-1})}^2-\left(\frac{2}{x{(x-1)}^2}\right))& =0\\ 0=0\end{array}$$

The equation is satisfied since both sides are zero. Therefore the first solution to the ode $z^{″}=rz$ is

$$\begin{array}{rl}{z}_1(x)& =p{e}^{\int \omega dx}\\ & ={\mathrm{e}}^{\int (\frac{1}{x}-\frac{1}{x-1})dx}\\ & =\frac{x}{x-1}\end{array}$$

The first solution to the original ode in $y$ is found from

$$y_1=z_1{e}^{\int -\frac{1}{2}\frac{B}{A}dx}$$

Since $B=0$ then the above reduces to

$$\begin{array}{rl}{y}_1& =z_1\\ & =\frac{x}{x-1}\end{array}$$

Which simplifies to

$$y_1=\frac{x}{x-1}$$

The second solution $y_2$ to the original ode is found using reduction of order

$$y_2=y_1\int \frac{e^{\int -\frac{B}{A}dx}}{y_1^2}dx$$

Since $B=0$ then the above becomes

$$\begin{array}{rl}{y}_2& =y_1\int \frac{1}{y_1^2}dx\\ & =\frac{x}{x-1}\int \frac{1}{\frac{x^2}{{(x-1)}^2}}dx\\ & =\frac{x}{x-1}(x-2\ln \left(x\right)-\frac{1}{x})\end{array}$$

Therefore the solution is

$$\begin{array}{rl}y& =c_1{y}_1+c_2{y}_2\\ & =c_1\left(\frac{x}{x-1}\right)+c_2\left(\frac{x}{x-1}(x-2\ln \left(x\right)-\frac{1}{x})\right)\end{array}$$

Will add steps showing solving for IC soon.

✓ Maple. Time used: 0.004 (sec). Leaf size: 27

ode:=x*(x-1)^2*diff(diff(y(x),x),x)-2*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{2\ln \left(x\right)c_{2}x-c_2{x}^2+c_{1}x+c_2}{x-1}$$

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
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 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
<- Kovacics algorithm successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & x{(x-1)}^2\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)-2y\left(x\right)=0\\ \bullet & & \text{Highest derivative means the order of the ODE is}2\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)\\ \bullet & & \text{Isolate 2nd derivative}\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)=\frac{2y\left(x\right)}{x{(x-1)}^2}\\ \bullet & & \text{Group terms with}y\left(x\right)\text{on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\\ & & \frac{d}{dx}\frac{d}{dx}y\left(x\right)-\frac{2y\left(x\right)}{x{(x-1)}^2}=0\\ ◻& & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & \circ & \text{Define functions}\\ & & [P_2\left(x\right)=0,P_3\left(x\right)=-\frac{2}{x{(x-1)}^2}]\\ & \circ & x\cdot P_2\left(x\right)\text{is analytic at}x=0\\ & & (x\cdot P_2\left(x\right))|{}_{x=0}=0\\ & \circ & x^2\cdot P_3\left(x\right)\text{is analytic at}x=0\\ & & (x^2\cdot P_3\left(x\right))|{}_{x=0}=0\\ & \circ & x=0\text{is a regular singular point}\\ & & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & & x_0=0\\ \bullet & & \text{Multiply by denominators}\\ & & x{(x-1)}^2\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)-2y\left(x\right)=0\\ \bullet & & \text{Assume series solution for}y\left(x\right)\\ & & y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^{k+r}\\ ◻& & \text{Rewrite ODE with series expansions}\\ & \circ & \text{Convert}{x}^m\cdot \left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)\text{to series expansion for}m=1..3\\ & & x^m\cdot \left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k(k+r)(k+r-1)x^{k+r-2+m}\\ & \circ & \text{Shift index using}k->k+2-m\\ & & x^m\cdot \left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=-2+m}}{a}_{k+2-m}(k+2-m+r)(k+1-m+r)x^{k+r}\\ & & \text{Rewrite ODE with series expansions}\\ & & a_{0}r(-1+r)x^{-1+r}+(a_1(1+r)r-2a_0(r^2-r+1))x^r+(\stackrel{\mathrm{\infty }}{\sum _{k=1}}(a_{k+1}(k+1+r)(k+r)-2a_k(k^2+2kr+r^2-k-r+1)+a_{k-1}(k+r-1)(k-2+r))x^{k+r})=0\\ \bullet & & a_0\text{cannot be 0 by assumption, giving the indicial equation}\\ & & r(-1+r)=0\\ \bullet & & \text{Values of r that satisfy the indicial equation}\\ & & r\in \{0,1\}\\ \bullet & & \text{Each term must be 0}\\ & & a_1(1+r)r-2a_0(r^2-r+1)=0\\ \bullet & & \text{Each term in the series must be 0, giving the recursion relation}\\ & & (-2a_k+a_{k-1}+a_{k+1})k^2+((-4a_k+2a_{k-1}+2a_{k+1})r+2a_k-3a_{k-1}+a_{k+1})k+(-2a_k+a_{k-1}+a_{k+1})r^2+(2a_k-3a_{k-1}+a_{k+1})r-2a_k+2a_{k-1}=0\\ \bullet & & \text{Shift index using}k->k+1\\ & & (-2a_{k+1}+a_k+a_{k+2}){(k+1)}^2+((-4a_{k+1}+2a_k+2a_{k+2})r+2a_{k+1}-3a_k+a_{k+2})(k+1)+(-2a_{k+1}+a_k+a_{k+2})r^2+(2a_{k+1}-3a_k+a_{k+2})r-2a_{k+1}+2a_k=0\\ \bullet & & \text{Recursion relation that defines series solution to ODE}\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+2kr{a}_k-4kr{a}_{k+1}+r^2{a}_k-2r^2{a}_{k+1}-k{a}_k-2k{a}_{k+1}-r{a}_k-2r{a}_{k+1}-2a_{k+1}}{k^2+2kr+r^2+3k+3r+2}\\ \bullet & & \text{Recursion relation for}r=0\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}-k{a}_k-2k{a}_{k+1}-2a_{k+1}}{k^2+3k+2}\\ \bullet & & \text{Solution for}r=0\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^k,a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}-k{a}_k-2k{a}_{k+1}-2a_{k+1}}{k^2+3k+2},-2a_0=0]\\ \bullet & & \text{Recursion relation for}r=1\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+k{a}_k-6k{a}_{k+1}-6a_{k+1}}{k^2+5k+6}\\ \bullet & & \text{Solution for}r=1\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^{k+1},a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+k{a}_k-6k{a}_{k+1}-6a_{k+1}}{k^2+5k+6},2a_1-2a_0=0]\\ \bullet & & \text{Combine solutions and rename parameters}\\ & & [y\left(x\right)=(\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^k)+(\stackrel{\mathrm{\infty }}{\sum _{k=0}}{b}_k{x}^{k+1}),a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}-k{a}_k-2k{a}_{k+1}-2a_{k+1}}{k^2+3k+2},-2a_0=0,b_{k+2}=-\frac{k^2{b}_k-2k^2{b}_{k+1}+k{b}_k-6k{b}_{k+1}-6b_{k+1}}{k^2+5k+6},2b_1-2b_0=0]\end{array}$$

✓ Mathematica. Time used: 0.068 (sec). Leaf size: 62

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

$$y(x)\to \exp ({\int }_1^x\frac{1}{K[1]-K[1{]}^2}dK[1])(c_2{\int }_1^x\exp (-2{\int }_1^{K[2]}\frac{1}{K[1]-K[1{]}^2}dK[1])dK[2]+c_1)$$

✗ Sympy

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

NotImplementedError : The given ODE x*(x - 1)**2*Derivative(y(x), (x, 2)) - 2*y(x) cannot be solved by the hypergeometric method