2.1.37 Problem 39

Internal problem ID [9207]
Book : Collection of Kovacic problems
Section : section 1
Problem number : 39
Date solved : Friday, April 25, 2025 at 05:59:50 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

Solve

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

Solved as second order ode using Kovacic algorithm

Time used: 0.210 (sec)

Writing the ode as

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

Comparing (1) and (2) shows that

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

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

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

Comparing the above to (5) shows that

$$\begin{array}{rl}s& =-1\\ t& =1\end{array}$$

Therefore eq. (4) becomes

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

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 2.37: Necessary conditions for each Kovacic case

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

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

There are no poles in $r$. Therefore the set of poles $\mathrm{\Gamma }$ is empty. Since there is no odd order pole larger than $2$ and the order at $\mathrm{\infty }$ is $0$ then the necessary conditions for case one are met. Therefore

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

Since $r=-1$ is not a function of $x$, then there is no need run Kovacic algorithm to obtain a solution for transformed ode $z^{″}=rz$ as one solution is

$$z_1(x)=\cos \left(x\right)$$

Using the above, the solution for the original ode can now be found. The first solution to the original ode in $y$ is found from

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

Which simplifies to

$$y_1=\sqrt{x}\cos \left(x\right)$$

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

$$\begin{array}{rl}{y}_2& =y_1\int \frac{e^{\int -\frac{-4x}{4x^2}dx}}{{\left(y_1\right)}^2}dx\\ & =y_1\int \frac{e^{\ln \left(x\right)}}{{\left(y_1\right)}^2}dx\\ & =y_1(\tan \left(x\right))\end{array}$$

Therefore the solution is

$$\begin{array}{rl}y& =c_1{y}_1+c_2{y}_2\\ & =c_1(\sqrt{x}\cos \left(x\right))+c_2(\sqrt{x}\cos \left(x\right)(\tan \left(x\right)))\end{array}$$

Will add steps showing solving for IC soon.

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

$$y=\sqrt{x}(c_1\sin \left(x\right)+c_2\cos \left(x\right))$$

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 
   Group is reducible or imprimitive 
<- Kovacics algorithm successful
 

Maple step by step

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

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 39

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

$$y(x)\to \frac{1}{2}{e}^{-ix}\sqrt{x}(2c_1-i{c}_2{e}^{2ix})$$

✓ Sympy. Time used: 0.212 (sec). Leaf size: 17

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

$$y\left(x\right)=x(C_1{J}_{\frac{1}{2}}\left(x\right)+C_2{Y}_{\frac{1}{2}}\left(x\right))$$