2.1.554 Problem 570

Internal problem ID [9724]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 570
Date solved : Friday, April 25, 2025 at 06:18:09 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

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

Solved as second order ode using Kovacic algorithm

Time used: 0.258 (sec)

Writing the ode as

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

Comparing (1) and (2) shows that

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

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

$$\begin{array}{}\text{(6)}& r& =\frac{5x^2+8x-16}{16{(x^2+2x)}^2}\end{array}$$

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =5x^2+8x-16\\ t& =16{(x^2+2x)}^2\end{array}$$

Therefore eq. (4) becomes

$$\begin{array}{}\text{(7)}& z^{″}(x)& =\left(\frac{5x^2+8x-16}{16{(x^2+2x)}^2}\right)z(x)\end{array}$$

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 2.554: 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 $t=16{(x^2+2x)}^2$. There is a pole at $x=0$ of order $2$. There is a pole at $x=-2$ 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

$$r=\frac{3}{8x}-\frac{1}{4x^2}-\frac{3}{16{(2+x)}^2}-\frac{3}{8(2+x)}$$

For the pole at $x=-2$ let $b$ be the coefficient of $\frac{1}{{(2+x)}^2}$ in the partial fractions decomposition of $r$ given above. Therefore $b=-\frac{3}{16}$. Hence

$$\begin{array}{rl}[\sqrt{r}{]}_c& =0\\ {\alpha }_c^{+}& =\frac{1}{2}+\sqrt{1+4b}& & =\frac{3}{4}\\ {\alpha }_c^{-}& =\frac{1}{2}-\sqrt{1+4b}& & =\frac{1}{4}\end{array}$$

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 $b=-\frac{1}{4}$. Hence

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

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

$$\begin{array}{rlrl}r& =\frac{s}{t}& & =\frac{5x^2+8x-16}{16{(x^2+2x)}^2}\end{array}$$

Since the $\text{gcd}(s,t)=1$. This gives $b=\frac{5}{16}$. Hence

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

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

$$r=\frac{5x^2+8x-16}{16{(x^2+2x)}^2}$$

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

Order of $r$ at $\mathrm{\infty }$	$[\sqrt{r}{]}_{\mathrm{\infty }}$	${\alpha }_{\mathrm{\infty }}^{+}$	${\alpha }_{\mathrm{\infty }}^{-}$
$2$	$0$	$\frac{5}{4}$	$-\frac{1}{4}$

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 ${\alpha }_{\mathrm{\infty }}^{+}=\frac{5}{4}$ then

$$\begin{array}{rl}d& ={\alpha }_{\mathrm{\infty }}^{+}-({\alpha }_{c_1}^{+}+{\alpha }_{c_2}^{+})\\ & =\frac{5}{4}-\left(\frac{5}{4}\right)\\ & =0\end{array}$$

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

$$\begin{array}{rl}\omega & =((+)[\sqrt{r}{]}_{c_1}+\frac{{\alpha }_{c_1}^{+}}{x-c_1})+((+)[\sqrt{r}{]}_{c_2}+\frac{{\alpha }_{c_2}^{+}}{x-c_2})+(+)[\sqrt{r}{]}_{\mathrm{\infty }}\\ & =\frac{3}{4(2+x)}+\frac{1}{2x}+\left(0\right)\\ & =\frac{3}{4(2+x)}+\frac{1}{2x}\\ & =\frac{5x+4}{4x(2+x)}\end{array}$$

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

$$\begin{array}{rl}\left(0\right)+2(\frac{3}{4(2+x)}+\frac{1}{2x})\left(0\right)+((-\frac{3}{4{(2+x)}^2}-\frac{1}{2x^2})+{(\frac{3}{4(2+x)}+\frac{1}{2x})}^2-\left(\frac{5x^2+8x-16}{16{(x^2+2x)}^2}\right))& =0\\ 0=0\end{array}$$

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

$$\begin{array}{rl}{z}_1(x)& =p{e}^{\int \omega dx}\\ & ={\mathrm{e}}^{\int (\frac{3}{4(2+x)}+\frac{1}{2x})dx}\\ & ={(2+x)}^{3/4}\sqrt{x}\end{array}$$

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

$$\begin{array}{rl}{y}_1& =z_1{e}^{\int -\frac{1}{2}\frac{B}{A}dx}\\ & =z_1{e}^{-\int \frac{1}{2}\frac{x^2}{2x^3+4x^2}dx}\\ & =z_1{e}^{-\frac{\ln (2+x)}{4}}\\ & =z_1\left(\frac{1}{{(2+x)}^{1/4}}\right)\end{array}$$

Which simplifies to

$$y_1=\sqrt{2+x}\sqrt{x}$$

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

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

Therefore the solution is

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

Will add steps showing solving for IC soon.

✓ Maple. Time used: 0.025 (sec). Leaf size: 50

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

$$y=c_1\sqrt{x(2+x)}+\frac{c_2((2+x)\mathrm{arctanh}\left(\frac{\sqrt{2+x}\sqrt{2}}{2}\right)-\sqrt{2+x}\sqrt{2})\sqrt{x}}{\sqrt{2+x}}$$

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}\\ & & 2x^2(2+x)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)+x^2\left(\frac{d}{dx}y\left(x\right)\right)+(1-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{(-1+x)y\left(x\right)}{2x^2(2+x)}-\frac{\frac{d}{dx}y\left(x\right)}{2(2+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{\frac{d}{dx}y\left(x\right)}{2(2+x)}-\frac{(-1+x)y\left(x\right)}{2x^2(2+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}{2(2+x)},P_3\left(x\right)=-\frac{-1+x}{2x^2(2+x)}]\\ & \circ & (2+x)\cdot P_2\left(x\right)\text{is analytic at}x=-2\\ & & ((2+x)\cdot P_2\left(x\right))|{}_{x=-2}=\frac{1}{2}\\ & \circ & {(2+x)}^2\cdot P_3\left(x\right)\text{is analytic at}x=-2\\ & & ({(2+x)}^2\cdot P_3\left(x\right))|{}_{x=-2}=0\\ & \circ & x=-2\text{is a regular singular point}\\ & & \text{Check to see if}{x}_0\text{is a regular singular point}\\ & & x_0=-2\\ \bullet & & \text{Multiply by denominators}\\ & & 2x^2(2+x)\left(\frac{d}{dx}\frac{d}{dx}y\left(x\right)\right)+x^2\left(\frac{d}{dx}y\left(x\right)\right)+(1-x)y\left(x\right)=0\\ \bullet & & \text{Change variables using}x=u-2\text{so that the regular singular point is at}u=0\\ & & (2u^3-8u^2+8u)\left(\frac{d}{du}\frac{d}{du}y\left(u\right)\right)+(u^2-4u+4)\left(\frac{d}{du}y\left(u\right)\right)+(3-u)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}\\ & & 4a_{0}r(-1+2r)u^{-1+r}+(4a_1(1+r)(1+2r)-a_0(8r^2-4r-3))u^r+(\stackrel{\mathrm{\infty }}{\sum _{k=1}}(4a_{k+1}(k+1+r)(2k+2r+1)-a_k(8k^2+16kr+8r^2-4k-4r-3)+a_{k-1}(2k+2r-1)(k-2+r))u^{k+r})=0\\ \bullet & & a_0\text{cannot be 0 by assumption, giving the indicial equation}\\ & & 4r(-1+2r)=0\\ \bullet & & \text{Values of r that satisfy the indicial equation}\\ & & r\in \{0,\frac{1}{2}\}\\ \bullet & & \text{Each term must be 0}\\ & & 4a_1(1+r)(1+2r)-a_0(8r^2-4r-3)=0\\ \bullet & & \text{Each term in the series must be 0, giving the recursion relation}\\ & & 2(-4a_k+a_{k-1}+4a_{k+1})k^2+(4(-4a_k+a_{k-1}+4a_{k+1})r+4a_k-5a_{k-1}+12a_{k+1})k+2(-4a_k+a_{k-1}+4a_{k+1})r^2+(4a_k-5a_{k-1}+12a_{k+1})r+3a_k+2a_{k-1}+4a_{k+1}=0\\ \bullet & & \text{Shift index using}k->k+1\\ & & 2(-4a_{k+1}+a_k+4a_{k+2}){(k+1)}^2+(4(-4a_{k+1}+a_k+4a_{k+2})r+4a_{k+1}-5a_k+12a_{k+2})(k+1)+2(-4a_{k+1}+a_k+4a_{k+2})r^2+(4a_{k+1}-5a_k+12a_{k+2})r+3a_{k+1}+2a_k+4a_{k+2}=0\\ \bullet & & \text{Recursion relation that defines series solution to ODE}\\ & & a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}+4kr{a}_k-16kr{a}_{k+1}+2r^2{a}_k-8r^2{a}_{k+1}-k{a}_k-12k{a}_{k+1}-r{a}_k-12r{a}_{k+1}-a_k-a_{k+1}}{4(2k^2+4kr+2r^2+7k+7r+6)}\\ \bullet & & \text{Recursion relation for}r=0\\ & & a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}-k{a}_k-12k{a}_{k+1}-a_k-a_{k+1}}{4(2k^2+7k+6)}\\ \bullet & & \text{Solution for}r=0\\ & & [y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{u}^k,a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}-k{a}_k-12k{a}_{k+1}-a_k-a_{k+1}}{4(2k^2+7k+6)},4a_1+3a_0=0]\\ \bullet & & \text{Revert the change of variables}u=2+x\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{(2+x)}^k,a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}-k{a}_k-12k{a}_{k+1}-a_k-a_{k+1}}{4(2k^2+7k+6)},4a_1+3a_0=0]\\ \bullet & & \text{Recursion relation for}r=\frac{1}{2}\\ & & a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}+k{a}_k-20k{a}_{k+1}-a_k-9a_{k+1}}{4(2k^2+9k+10)}\\ \bullet & & \text{Solution for}r=\frac{1}{2}\\ & & [y\left(u\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{u}^{k+\frac{1}{2}},a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}+k{a}_k-20k{a}_{k+1}-a_k-9a_{k+1}}{4(2k^2+9k+10)},12a_1+3a_0=0]\\ \bullet & & \text{Revert the change of variables}u=2+x\\ & & [y\left(x\right)=\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{(2+x)}^{k+\frac{1}{2}},a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}+k{a}_k-20k{a}_{k+1}-a_k-9a_{k+1}}{4(2k^2+9k+10)},12a_1+3a_0=0]\\ \bullet & & \text{Combine solutions and rename parameters}\\ & & [y\left(x\right)=(\stackrel{\mathrm{\infty }}{\sum _{k=0}}{a}_k{(2+x)}^k)+(\stackrel{\mathrm{\infty }}{\sum _{k=0}}{b}_k{(2+x)}^{k+\frac{1}{2}}),a_{k+2}=-\frac{2k^2{a}_k-8k^2{a}_{k+1}-k{a}_k-12k{a}_{k+1}-a_k-a_{k+1}}{4(2k^2+7k+6)},4a_1+3a_0=0,b_{k+2}=-\frac{2k^2{b}_k-8k^2{b}_{k+1}+k{b}_k-20k{b}_{k+1}-b_k-9b_{k+1}}{4(2k^2+9k+10)},12b_1+3b_0=0]\end{array}$$

✓ Mathematica. Time used: 0.51 (sec). Leaf size: 92

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

$$y(x)\to \frac{\exp ({\int }_1^x\frac{5K[1]+4}{4K[1{]}^2+8K[1]}dK[1])(c_2{\int }_1^x\exp (-2{\int }_1^{K[2]}\frac{5K[1]+4}{4K[1{]}^2+8K[1]}dK[1])dK[2]+c_1)}{\sqrt[4]{2}\sqrt[4]{x+2}}$$

✓ Sympy. Time used: 1.095 (sec). Leaf size: 97

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

$$y\left(x\right)=\sqrt{x}(C_1+C_2\left(\{\begin{array}{ll}2\sqrt{\frac{x}{x+2}-1}+2\mathrm{asin}\left(\frac{1}{\sqrt{\frac{x}{x+2}}}\right)& \text{for}\left|\frac{x}{x+2}\right|>1\\ 2i\sqrt{-\frac{x}{x+2}+1}+i\log \left(\frac{x}{x+2}\right)-2i\log (\sqrt{-\frac{x}{x+2}+1}+1)& \text{otherwise}\end{array}\right))\sqrt{x+2}{}_2{F}_1\left(\begin{array}{c}-\frac{1}{2},0\\ 1\end{array}\right|\frac{x}{x+2})$$