2.1.541 Problem 557

Internal problem ID [9711]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 557
Date solved : Friday, April 25, 2025 at 06:17:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

$$\begin{array}{rl}{x}^2(x^2-2x+1)y^{\prime \prime }-x(3+x)y^{\prime }+(4+x)y& =0\end{array}$$

Solved as second order ode using Kovacic algorithm

Time used: 0.408 (sec)

Writing the ode as

$$\begin{array}{}\text{(1)}& x^2{(x-1)}^2{y}^{\prime \prime }+(-x^2-3x)y^{\prime }+(4+x)y& =0\text{(2)}& A{y}^{\prime \prime }+B{y}^{\prime }+Cy& =0\end{array}$$

Comparing (1) and (2) shows that

$$\begin{array}{rl}A& =x^2{(x-1)}^2\\ \text{(3)}& B& =-x^2-3xC& =4+x\end{array}$$

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

$$\begin{array}{}\text{(6)}& r& =\frac{7x^2+10x-1}{4x^2{(x-1)}^4}\end{array}$$

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =7x^2+10x-1\\ t& =4x^2{(x-1)}^4\end{array}$$

Therefore eq. (4) becomes

$$\begin{array}{}\text{(7)}& z^{″}(x)& =\left(\frac{7x^2+10x-1}{4x^2{(x-1)}^4}\right)z(x)\end{array}$$

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 2.541: Necessary conditions for each Kovacic case

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

$$\begin{array}{rl}O\left(\mathrm{\infty }\right)& =\text{deg}(t)-\text{deg}(s)\\ & =6-2\\ & =4\end{array}$$

The poles of $r$ in eq. (7) and the order of each pole are determined by solving for the roots of $t=4x^2{(x-1)}^4$. There is a pole at $x=0$ of order $2$. There is a pole at $x=1$ of order $4$. Since there is no odd order pole larger than $2$ and the order at $\mathrm{\infty }$ is $4$ 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. Therefore

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

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

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

$$r=\frac{7}{4{(x-1)}^2}-\frac{3}{2(x-1)}-\frac{2}{{(x-1)}^3}+\frac{4}{{(x-1)}^4}+\frac{3}{2x}-\frac{1}{4x^2}$$

For the pole at $x=0$ let $b$ be the coefficient of $\frac{1}{x^2}$ in the partial fractions decomposition of $r$ given above. Therefore $b=-\frac{1}{4}$. Hence

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

Looking at higher order poles of order $2v$≥$4$ (must be even order for case one).Then for each pole $c$, $[\sqrt{r}{]}_c$ is the sum of terms $\frac{1}{(x-c{)}^i}$ for $2\le i\le v$ in the Laurent series expansion of $\sqrt{r}$ expanded around each pole $c$. Hence

$$\begin{array}{}\text{(1B)}& [\sqrt{r}{]}_c& =\sum _2^v\frac{a_i}{(x-c{)}^i}\end{array}$$

Let $a$ be the coefficient of the term $\frac{1}{(x-c{)}^v}$ in the above where $v$ is the pole order divided by 2. Let $b$ be the coefficient of $\frac{1}{(x-c{)}^{v+1}}$ in $r$ minus the coefficient of $\frac{1}{(x-c{)}^{v+1}}$ in $[\sqrt{r}{]}_c$. Then

$$\begin{array}{rl}{\alpha }_c^{+}& =\frac{1}{2}(\frac{b}{a}+v)\\ {\alpha }_c^{-}& =\frac{1}{2}(-\frac{b}{a}+v)\end{array}$$

The partial fraction decomposition of $r$ is

$$r=\frac{7}{4{(x-1)}^2}-\frac{3}{2(x-1)}-\frac{2}{{(x-1)}^3}+\frac{4}{{(x-1)}^4}+\frac{3}{2x}-\frac{1}{4x^2}$$

There is pole in $r$ at $x=1$ of order $4$, hence $v=2$. Expanding $\sqrt{r}$ as Laurent series about this pole $c=1$ gives

$$\begin{array}{}\text{(2B)}& [\sqrt{r}{]}_c\approx \frac{2}{{(x-1)}^2}-\frac{1}{2(x-1)}+\frac{21}{32}-\frac{9x}{32}+\frac{53{(x-1)}^2}{256}-\frac{149{(x-1)}^3}{1024}+\dots \end{array}$$

Using eq. (1B), taking the sum up to $v=2$ the above becomes

$$\begin{array}{}\text{(3B)}& [\sqrt{r}{]}_c=\frac{2}{{(x-1)}^2}\end{array}$$

The above shows that the coefficient of $\frac{1}{(x-1{)}^2}$ is

$$a=2$$

Now we need to find $b$. let $b$ be the coefficient of the term $\frac{1}{(x-c{)}^{v+1}}$ in $r$ minus the coefficient of the same term but in the sum $[\sqrt{r}{]}_c$ found in eq. (3B). Here $c$ is current pole which is $c=1$. This term becomes $\frac{1}{{(x-1)}^3}$. The coefficient of this term in the sum $[\sqrt{r}{]}_c$ is seen to be $0$ and the coefficient of this term $r$ is found from the partial fraction decomposition from above to be $-2$. Therefore

$$\begin{array}{rl}b& =(-2)-(0)\\ & =-2\end{array}$$

Hence

$$\begin{array}{rl}[\sqrt{r}{]}_c& =\frac{2}{{(x-1)}^2}\\ {\alpha }_c^{+}& =\frac{1}{2}(\frac{b}{a}+v)& & =\frac{1}{2}(\frac{-2}{2}+2)& & =\frac{1}{2}\\ {\alpha }_c^{-}& =\frac{1}{2}(-\frac{b}{a}+v)& & =\frac{1}{2}(-\frac{-2}{2}+2)& & =\frac{3}{2}\end{array}$$

Since the order of $r$ at $\mathrm{\infty }$ is $4>2$ 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

$$r=\frac{7x^2+10x-1}{4x^2{(x-1)}^4}$$

pole $c$ location	pole order	$[\sqrt{r}{]}_c$	${\alpha }_c^{+}$	${\alpha }_c^{-}$
$0$	$2$	$0$	$\frac{1}{2}$	$\frac{1}{2}$
$1$	$4$	$\frac{2}{{(x-1)}^2}$	$\frac{1}{2}$	$\frac{3}{2}$

Order of $r$ at $\mathrm{\infty }$	$[\sqrt{r}{]}_{\mathrm{\infty }}$	${\alpha }_{\mathrm{\infty }}^{+}$	${\alpha }_{\mathrm{\infty }}^{-}$
$4$	$0$	$0$	$1$

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 ${\alpha }_{\mathrm{\infty }}^{-}=1$ then

$$\begin{array}{rl}d& ={\alpha }_{\mathrm{\infty }}^{-}-({\alpha }_{c_1}^{+}+{\alpha }_{c_2}^{+})\\ & =1-\left(1\right)\\ & =0\end{array}$$

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

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

The above gives

$$\begin{array}{rl}\omega & =((+)[\sqrt{r}{]}_{c_1}+\frac{{\alpha }_{c_1}^{+}}{x-c_1})+((+)[\sqrt{r}{]}_{c_2}+\frac{{\alpha }_{c_2}^{+}}{x-c_2})+(-)[\sqrt{r}{]}_{\mathrm{\infty }}\\ & =\frac{1}{2x}+\frac{2}{{(x-1)}^2}+\frac{1}{2x-2}+(-)\left(0\right)\\ & =\frac{1}{2x}+\frac{2}{{(x-1)}^2}+\frac{1}{2x-2}\\ & =\frac{2x^2+x+1}{2x{(x-1)}^2}\end{array}$$

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

$$\begin{array}{rl}\left(0\right)+2(\frac{1}{2x}+\frac{2}{{(x-1)}^2}+\frac{1}{2x-2})\left(0\right)+((-\frac{1}{2x^2}-\frac{4}{{(x-1)}^3}-\frac{1}{2{(x-1)}^2})+{(\frac{1}{2x}+\frac{2}{{(x-1)}^2}+\frac{1}{2x-2})}^2-\left(\frac{7x^2+10x-1}{4x^2{(x-1)}^4}\right))& =0\\ 0=0\end{array}$$

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

$$\begin{array}{rl}{z}_1(x)& =p{e}^{\int \omega dx}\\ & ={\mathrm{e}}^{\int (\frac{1}{2x}+\frac{2}{{(x-1)}^2}+\frac{1}{2x-2})dx}\\ & =\sqrt{x-1}\sqrt{x}{\mathrm{e}}^{-\frac{2}{x-1}}\end{array}$$

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

$$\begin{array}{rl}{y}_1& =z_1{e}^{\int -\frac{1}{2}\frac{B}{A}dx}\\ & =z_1{e}^{-\int \frac{1}{2}\frac{-x^2-3x}{x^2{(x-1)}^2}dx}\\ & =z_1{e}^{-\frac{2}{x-1}-\frac{3\ln (x-1)}{2}+\frac{3\ln \left(x\right)}{2}}\\ & =z_1\left(\frac{x^{3/2}{\mathrm{e}}^{-\frac{2}{x-1}}}{{(x-1)}^{3/2}}\right)\end{array}$$

Which simplifies to

$$y_1=\frac{x^{3/2}{\mathrm{e}}^{-\frac{4}{x-1}}\sqrt{x(x-1)}}{{(x-1)}^{3/2}}$$

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

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

Substituting gives

$$\begin{array}{rl}{y}_2& =y_1\int \frac{e^{\int -\frac{-x^2-3x}{x^2{(x-1)}^2}dx}}{{\left(y_1\right)}^2}dx\\ & =y_1\int \frac{e^{-\frac{4}{x-1}-3\ln (x-1)+3\ln \left(x\right)}}{{\left(y_1\right)}^2}dx\\ & =y_1\left({\mathrm{e}}^{-4}{\mathrm{Ei}}_1(-\frac{4}{x-1}-4)\right)\end{array}$$

Therefore the solution is

$$\begin{array}{rl}y& =c_1{y}_1+c_2{y}_2\\ & =c_1\left(\frac{x^{3/2}{\mathrm{e}}^{-\frac{4}{x-1}}\sqrt{x(x-1)}}{{(x-1)}^{3/2}}\right)+c_2\left(\frac{x^{3/2}{\mathrm{e}}^{-\frac{4}{x-1}}\sqrt{x(x-1)}}{{(x-1)}^{3/2}}\left({\mathrm{e}}^{-4}{\mathrm{Ei}}_1(-\frac{4}{x-1}-4)\right)\right)\end{array}$$

Will add steps showing solving for IC soon.

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

ode:=x^2*(x^2-2*x+1)*diff(diff(y(x),x),x)-x*(x+3)*diff(y(x),x)+(x+4)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{x^2({\mathrm{Ei}}_1(-\frac{4x}{x-1}){\mathrm{e}}^{-\frac{4x}{x-1}}{c}_2+{\mathrm{e}}^{-\frac{4}{x-1}}{c}_1)}{x-1}$$

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^2(x^2-2x+1)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)-x(3+x)\left(\frac{d}{dx}y\left(x\right)\right)+(4+x)y\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{(4+x)y\left(x\right)}{x^2(x^2-2x+1)}+\frac{(3+x)\left(\frac{d}{dx}y\left(x\right)\right)}{x(x^2-2x+1)}\\ \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{(3+x)\left(\frac{d}{dx}y\left(x\right)\right)}{x(x^2-2x+1)}+\frac{(4+x)y\left(x\right)}{x^2(x^2-2x+1)}=0\\ ◻& & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & \circ & \text{Define functions}\\ & & [P_2\left(x\right)=-\frac{3+x}{x(x^2-2x+1)},P_3\left(x\right)=\frac{4+x}{x^2(x^2-2x+1)}]\\ & \circ & x\cdot P_2\left(x\right)\text{is analytic at}x=0\\ & & (x\cdot P_2\left(x\right))|{}_{x=0}=-3\\ & \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}=4\\ & \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^2(x^2-2x+1)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)-x(3+x)\left(\frac{d}{dx}y\left(x\right)\right)+(4+x)y\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 y\left(x\right)\text{to series expansion for}m=0..1\\ & & x^m\cdot y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^{k+r+m}\\ & \circ & \text{Shift index using}k->k-m\\ & & x^m\cdot y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=m}}{a}_{k-m}{x}^{k+r}\\ & \circ & \text{Convert}{x}^m\cdot \left(\frac{d}{dx}y\left(x\right)\right)\text{to series expansion for}m=1..2\\ & & x^m\cdot \left(\frac{d}{dx}y\left(x\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k(k+r)x^{k+r-1+m}\\ & \circ & \text{Shift index using}k->k+1-m\\ & & x^m\cdot \left(\frac{d}{dx}y\left(x\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=-1+m}}{a}_{k+1-m}(k+1-m+r)x^{k+r}\\ & \circ & \text{Convert}{x}^m\cdot \left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)\text{to series expansion for}m=2..4\\ & & 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{(-2+r)}^2{x}^r+(a_1{(-1+r)}^2-a_0(1+2r)(-1+r))x^{1+r}+(\stackrel{\mathrm{\infty }}{\sum _{k=2}}(a_k{(k+r-2)}^2-a_{k-1}(2k-1+2r)(k+r-2)+a_{k-2}(k+r-2)(k-3+r))x^{k+r})=0\\ \bullet & & a_0\text{cannot be 0 by assumption, giving the indicial equation}\\ & & {(-2+r)}^2=0\\ \bullet & & \text{Values of r that satisfy the indicial equation}\\ & & r=2\\ \bullet & & \text{Each term must be 0}\\ & & a_1{(-1+r)}^2-a_0(1+2r)(-1+r)=0\\ \bullet & & \text{Solve for the dependent coefficient(s)}\\ & & a_1=\frac{a_0(1+2r)}{-1+r}\\ \bullet & & \text{Each term in the series must be 0, giving the recursion relation}\\ & & (k+r-2)((a_k+a_{k-2}-2a_{k-1})k+(a_k+a_{k-2}-2a_{k-1})r-2a_k-3a_{k-2}+a_{k-1})=0\\ \bullet & & \text{Shift index using}k->k+2\\ & & (k+r)((a_{k+2}+a_k-2a_{k+1})(k+2)+(a_{k+2}+a_k-2a_{k+1})r-2a_{k+2}-3a_k+a_{k+1})=0\\ \bullet & & \text{Recursion relation that defines series solution to ODE}\\ & & a_{k+2}=-\frac{k{a}_k-2k{a}_{k+1}+r{a}_k-2r{a}_{k+1}-a_k-3a_{k+1}}{k+r}\\ \bullet & & \text{Recursion relation for}r=2\\ & & a_{k+2}=-\frac{k{a}_k-2k{a}_{k+1}+a_k-7a_{k+1}}{k+2}\\ \bullet & & \text{Solution for}r=2\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{x}^{k+2},a_{k+2}=-\frac{k{a}_k-2k{a}_{k+1}+a_k-7a_{k+1}}{k+2},a_1=5a_0]\end{array}$$

✓ Mathematica. Time used: 0.285 (sec). Leaf size: 116

ode=x^2*(1-2*x+x^2)*D[y[x],{x,2}]-x*(3+x)*D[y[x],x]+(4+x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to \exp ({\int }_1^x\frac{2K[1{]}^2+K[1]+1}{2(K[1]-1{)}^{2}K[1]}dK[1]-\frac{1}{2}{\int }_1^x-\frac{K[2]+3}{(K[2]-1{)}^{2}K[2]}dK[2])(c_2{\int }_1^x\exp (-2{\int }_1^{K[3]}\frac{2K[1{]}^2+K[1]+1}{2(K[1]-1{)}^{2}K[1]}dK[1])dK[3]+c_1)$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x**2 - 2*x + 1)*Derivative(y(x), (x, 2)) - x*(x + 3)*Derivative(y(x), x) + (x + 4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False