2.1.597 Problem 613

Internal problem ID [9767]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 613
Date solved : Friday, April 25, 2025 at 06:19:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

$$\begin{array}{rl}{x}^2(1+x)y^{\prime \prime }+x(1-10x)y^{\prime }-(9-10x)y& =0\end{array}$$

Solved as second order ode using Kovacic algorithm

Time used: 0.305 (sec)

Writing the ode as

$$\begin{array}{}\text{(1)}& x^2(1+x)y^{\prime \prime }+(-10x^2+x)y^{\prime }+(10x-9)y& =0\text{(2)}& A{y}^{\prime \prime }+B{y}^{\prime }+Cy& =0\end{array}$$

Comparing (1) and (2) shows that

$$\begin{array}{rl}A& =x^2(1+x)\\ \text{(3)}& B& =-10x^2+xC& =10x-9\end{array}$$

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

$$\begin{array}{}\text{(6)}& r& =\frac{80x^2-28x+35}{4{(x^2+x)}^2}\end{array}$$

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =80x^2-28x+35\\ t& =4{(x^2+x)}^2\end{array}$$

Therefore eq. (4) becomes

$$\begin{array}{}\text{(7)}& z^{″}(x)& =\left(\frac{80x^2-28x+35}{4{(x^2+x)}^2}\right)z(x)\end{array}$$

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 2.597: Necessary conditions for each Kovacic case

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

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

The poles of $r$ in eq. (7) and the order of each pole are determined by solving for the roots of $t=4{(x^2+x)}^2$. There is a pole at $x=0$ of order $2$. 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 $2$ 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 $2$ 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 2. The partial fractions decomposition of $r$ is

$$r=-\frac{49}{2x}+\frac{143}{4{(1+x)}^2}+\frac{49}{2(1+x)}+\frac{35}{4x^2}$$

For the pole at $x=-1$ let $b$ be the coefficient of $\frac{1}{{(1+x)}^2}$ in the partial fractions decomposition of $r$ given above. Therefore $b=\frac{143}{4}$. Hence

$$\begin{array}{rl}[\sqrt{r}{]}_c& =0\\ {\alpha }_c^{+}& =\frac{1}{2}+\sqrt{1+4b}& & =\frac{13}{2}\\ {\alpha }_c^{-}& =\frac{1}{2}-\sqrt{1+4b}& & =-\frac{11}{2}\end{array}$$

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 $b=\frac{35}{4}$. Hence

$$\begin{array}{rl}[\sqrt{r}{]}_c& =0\\ {\alpha }_c^{+}& =\frac{1}{2}+\sqrt{1+4b}& & =\frac{7}{2}\\ {\alpha }_c^{-}& =\frac{1}{2}-\sqrt{1+4b}& & =-\frac{5}{2}\end{array}$$

Since the order of $r$ at $\mathrm{\infty }$ is 2 then $[\sqrt{r}{]}_{\mathrm{\infty }}=0$. Let $b$ be the coefficient of $\frac{1}{x^2}$ in the Laurent series expansion of $r$ at $\mathrm{\infty }$. which can be found by dividing the leading coefficient of $s$ by the leading coefficient of $t$ from

$$\begin{array}{rlrl}r& =\frac{s}{t}& & =\frac{80x^2-28x+35}{4{(x^2+x)}^2}\end{array}$$

Since the $\text{gcd}(s,t)=1$. This gives $b=20$. Hence

$$\begin{array}{rl}[\sqrt{r}{]}_{\mathrm{\infty }}& =0\\ {\alpha }_{\mathrm{\infty }}^{+}& =\frac{1}{2}+\sqrt{1+4b}& & =5\\ {\alpha }_{\mathrm{\infty }}^{-}& =\frac{1}{2}-\sqrt{1+4b}& & =-4\end{array}$$

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

$$r=\frac{80x^2-28x+35}{4{(x^2+x)}^2}$$

pole $c$ location	pole order	$[\sqrt{r}{]}_c$	${\alpha }_c^{+}$	${\alpha }_c^{-}$
$-1$	$2$	$0$	$\frac{13}{2}$	$-\frac{11}{2}$
$0$	$2$	$0$	$\frac{7}{2}$	$-\frac{5}{2}$

Order of $r$ at $\mathrm{\infty }$	$[\sqrt{r}{]}_{\mathrm{\infty }}$	${\alpha }_{\mathrm{\infty }}^{+}$	${\alpha }_{\mathrm{\infty }}^{-}$
$2$	$0$	$5$	$-4$

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 ${\alpha }_{\mathrm{\infty }}^{+}=5$ then

$$\begin{array}{rl}d& ={\alpha }_{\mathrm{\infty }}^{+}-({\alpha }_{c_1}^{+}+{\alpha }_{c_2}^{-})\\ & =5-\left(4\right)\\ & =1\end{array}$$

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

$$\begin{array}{rl}\omega & =((+)[\sqrt{r}{]}_{c_1}+\frac{{\alpha }_{c_1}^{+}}{x-c_1})+((-)[\sqrt{r}{]}_{c_2}+\frac{{\alpha }_{c_2}^{-}}{x-c_2})+(+)[\sqrt{r}{]}_{\mathrm{\infty }}\\ & =\frac{13}{2(1+x)}-\frac{5}{2x}+\left(0\right)\\ & =\frac{13}{2(1+x)}-\frac{5}{2x}\\ & =\frac{8x-5}{2x(1+x)}\end{array}$$

Now that $\omega$ is determined, the next step is find a corresponding minimal polynomial $p(x)$ of degree $d=1$ 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

$$\begin{array}{}\text{(2A)}& p(x)& =x+a_0\end{array}$$

Substituting the above in eq. (1A) gives

$$\begin{array}{rl}\left(0\right)+2(\frac{13}{2(1+x)}-\frac{5}{2x})\left(1\right)+((-\frac{13}{2{(1+x)}^2}+\frac{5}{2x^2})+{(\frac{13}{2(1+x)}-\frac{5}{2x})}^2-\left(\frac{80x^2-28x+35}{4{(x^2+x)}^2}\right))& =0\\ \frac{-5-8a_0}{x(1+x)}=0\end{array}$$

Solving for the coefficients $a_i$ in the above using method of undetermined coefficients gives

$$\{a_0=-\frac{5}{8}\}$$

Substituting these coefficients in $p(x)$ in eq. (2A) results in

$$\begin{array}{rl}p(x)& =x-\frac{5}{8}\end{array}$$

Therefore the first solution to the ode $z^{″}=rz$ is

$$\begin{array}{rl}{z}_1(x)& =p{e}^{\int \omega dx}\\ & =(x-\frac{5}{8}){\mathrm{e}}^{\int (\frac{13}{2(1+x)}-\frac{5}{2x})dx}\\ & =(x-\frac{5}{8}){\mathrm{e}}^{\frac{13\ln (1+x)}{2}-\frac{5\ln \left(x\right)}{2}}\\ & =\frac{(x-\frac{5}{8}){(1+x)}^{13/2}}{x^{5/2}}\end{array}$$

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

$$\begin{array}{rl}{y}_1& =z_1{e}^{\int -\frac{1}{2}\frac{B}{A}dx}\\ & =z_1{e}^{-\int \frac{1}{2}\frac{-10x^2+x}{x^2(1+x)}dx}\\ & =z_1{e}^{\frac{11\ln (1+x)}{2}-\frac{\ln \left(x\right)}{2}}\\ & =z_1\left(\frac{{(1+x)}^{11/2}}{\sqrt{x}}\right)\end{array}$$

Which simplifies to

$$y_1=\frac{{(1+x)}^{12}(x-\frac{5}{8})}{x^3}$$

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

$$\begin{array}{rl}{y}_2& =y_1\int \frac{e^{\int -\frac{-10x^2+x}{x^2(1+x)}dx}}{{\left(y_1\right)}^2}dx\\ & =y_1\int \frac{e^{11\ln (1+x)-\ln \left(x\right)}}{{\left(y_1\right)}^2}dx\\ & =y_1(-\frac{8{\mathrm{e}}^{11\ln (1+x)-\ln \left(x\right)}x(715x^4+572x^3+234x^2+52x+5)}{6435(8x-5){(1+x)}^{23}})\end{array}$$

Therefore the solution is

$$\begin{array}{rl}y& =c_1{y}_1+c_2{y}_2\\ & =c_1\left(\frac{{(1+x)}^{12}(x-\frac{5}{8})}{x^3}\right)+c_2\left(\frac{{(1+x)}^{12}(x-\frac{5}{8})}{x^3}(-\frac{8{\mathrm{e}}^{11\ln (1+x)-\ln \left(x\right)}x(715x^4+572x^3+234x^2+52x+5)}{6435(8x-5){(1+x)}^{23}})\right)\end{array}$$

Will add steps showing solving for IC soon.

✓ Maple. Time used: 0.004 (sec). Leaf size: 82

ode:=x^2*(1+x)*diff(diff(y(x),x),x)+x*(1-10*x)*diff(y(x),x)-(9-10*x)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

$$y=\frac{8c_2{x}^{13}+91c_2{x}^{12}+468c_2{x}^{11}+1430c_2{x}^{10}+2860c_2{x}^9+3861c_2{x}^8+3432c_2{x}^7+1716c_2{x}^6+715c_1{x}^4+572c_1{x}^3+234c_1{x}^2+52c_{1}x+5c_1}{x^3}$$

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) 
<- Kovacics algorithm successful
 

Maple step by step

$$\begin{array}{lll}& & \text{Let’s solve}\\ & & x^2(1+x)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)+x(1-10x)\left(\frac{d}{dx}y\left(x\right)\right)-(9-10x)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{(-9+10x)y\left(x\right)}{x^2(1+x)}+\frac{(-1+10x)\left(\frac{d}{dx}y\left(x\right)\right)}{x(1+x)}\\ \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{(-1+10x)\left(\frac{d}{dx}y\left(x\right)\right)}{x(1+x)}+\frac{(-9+10x)y\left(x\right)}{x^2(1+x)}=0\\ ◻& & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & \circ & \text{Define functions}\\ & & [P_2\left(x\right)=-\frac{-1+10x}{x(1+x)},P_3\left(x\right)=\frac{-9+10x}{x^2(1+x)}]\\ & \circ & (1+x)\cdot P_2\left(x\right)\text{is analytic at}x=-1\\ & & ((1+x)\cdot P_2\left(x\right))|{}_{x=-1}=-11\\ & \circ & {(1+x)}^2\cdot P_3\left(x\right)\text{is analytic at}x=-1\\ & & ({(1+x)}^2\cdot P_3\left(x\right))|{}_{x=-1}=0\\ & \circ & x=-1\text{is a regular singular point}\\ & & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & & x_0=-1\\ \bullet & & \text{Multiply by denominators}\\ & & x^2(1+x)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)-x(-1+10x)\left(\frac{d}{dx}y\left(x\right)\right)+(-9+10x)y\left(x\right)=0\\ \bullet & & \text{Change variables using}x=u-1\text{so that the regular singular point is at}u=0\\ & & (u^3-2u^2+u)\left(\frac{d}{du}\frac{d}{du}y\left(u\right)\right)+(-10u^2+21u-11)\left(\frac{d}{du}y\left(u\right)\right)+(-19+10u)y\left(u\right)=0\\ \bullet & & \text{Assume series solution for}y\left(u\right)\\ & & y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{u}^{k+r}\\ ◻& & \text{Rewrite ODE with series expansions}\\ & \circ & \text{Convert}{u}^m\cdot y\left(u\right)\text{to series expansion for}m=0..1\\ & & u^m\cdot y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{u}^{k+r+m}\\ & \circ & \text{Shift index using}k->k-m\\ & & u^m\cdot y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=m}}{a}_{k-m}{u}^{k+r}\\ & \circ & \text{Convert}{u}^m\cdot \left(\frac{d}{du}y\left(u\right)\right)\text{to series expansion for}m=0..2\\ & & u^m\cdot \left(\frac{d}{du}y\left(u\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k(k+r)u^{k+r-1+m}\\ & \circ & \text{Shift index using}k->k+1-m\\ & & u^m\cdot \left(\frac{d}{du}y\left(u\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=-1+m}}{a}_{k+1-m}(k+1-m+r)u^{k+r}\\ & \circ & \text{Convert}{u}^m\cdot \left(\frac{d}{du}\frac{d}{du}y\left(u\right)\right)\text{to series expansion for}m=1..3\\ & & u^m\cdot \left(\frac{d}{du}\frac{d}{du}y\left(u\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k(k+r)(k+r-1)u^{k+r-2+m}\\ & \circ & \text{Shift index using}k->k+2-m\\ & & u^m\cdot \left(\frac{d}{du}\frac{d}{du}y\left(u\right)\right)=\stackrel{\mathrm{\infty }}{\sum _{k=-2+m}}{a}_{k+2-m}(k+2-m+r)(k+1-m+r)u^{k+r}\\ & & \text{Rewrite ODE with series expansions}\\ & & a_{0}r(-12+r)u^{-1+r}+(a_1(1+r)(-11+r)-a_0(2r^2-23r+19))u^r+(\stackrel{\mathrm{\infty }}{\sum _{k=1}}(a_{k+1}(k+1+r)(k-11+r)-a_k(2k^2+4kr+2r^2-23k-23r+19)+a_{k-1}(k-2+r)(k-11+r))u^{k+r})=0\\ \bullet & & a_0\text{cannot be 0 by assumption, giving the indicial equation}\\ & & r(-12+r)=0\\ \bullet & & \text{Values of r that satisfy the indicial equation}\\ & & r\in \{0,12\}\\ \bullet & & \text{Each term must be 0}\\ & & a_1(1+r)(-11+r)-a_0(2r^2-23r+19)=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+23a_k-13a_{k-1}-10a_{k+1})k+(-2a_k+a_{k-1}+a_{k+1})r^2+(23a_k-13a_{k-1}-10a_{k+1})r-19a_k+22a_{k-1}-11a_{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+23a_{k+1}-13a_k-10a_{k+2})(k+1)+(-2a_{k+1}+a_k+a_{k+2})r^2+(23a_{k+1}-13a_k-10a_{k+2})r-19a_{k+1}+22a_k-11a_{k+2}=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}-11k{a}_k+19k{a}_{k+1}-11r{a}_k+19r{a}_{k+1}+10a_k+2a_{k+1}}{k^2+2kr+r^2-8k-8r-20}\\ \bullet & & \text{Recursion relation for}r=0\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}-11k{a}_k+19k{a}_{k+1}+10a_k+2a_{k+1}}{k^2-8k-20}\\ \bullet & & \text{Series not valid for}r=0\text{, division by}0\text{in the recursion relation at}k=10\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}-11k{a}_k+19k{a}_{k+1}+10a_k+2a_{k+1}}{k^2-8k-20}\\ \bullet & & \text{Recursion relation for}r=12\\ & & a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+13k{a}_k-29k{a}_{k+1}+22a_k-58a_{k+1}}{k^2+16k+28}\\ \bullet & & \text{Solution for}r=12\\ & & [y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{u}^{k+12},a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+13k{a}_k-29k{a}_{k+1}+22a_k-58a_{k+1}}{k^2+16k+28},13a_1-31a_0=0]\\ \bullet & & \text{Revert the change of variables}u=1+x\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{(1+x)}^{k+12},a_{k+2}=-\frac{k^2{a}_k-2k^2{a}_{k+1}+13k{a}_k-29k{a}_{k+1}+22a_k-58a_{k+1}}{k^2+16k+28},13a_1-31a_0=0]\end{array}$$

✓ Mathematica. Time used: 0.472 (sec). Leaf size: 123

ode=x^2*(1+x)*D[y[x],{x,2}]+x*(1-10*x)*D[y[x],x]-(9-10*x)*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

$$y(x)\to \frac{1}{8}(8x-5)\exp ({\int }_1^x\frac{8K[1]-5}{2K[1](K[1]+1)}dK[1]-\frac{1}{2}{\int }_1^x(\frac{1}{K[2]}-\frac{11}{K[2]+1})dK[2])(c_2{\int }_1^x\frac{64\exp (-2{\int }_1^{K[3]}\frac{8K[1]-5}{2K[1](K[1]+1)}dK[1])}{(5-8K[3]{)}^2}dK[3]+c_1)$$

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(1 - 10*x)*Derivative(y(x), x) - (9 - 10*x)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False