1.33 problem 34
Internal
problem
ID
[8171]
Book
:
Collection
of
Kovacic
problems
Section
:
section
1
Problem
number
:
34
Date
solved
:
Monday, October 21, 2024 at 04:54:50 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
Solve
\begin{align*} \left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }-\left (2 x +3\right ) y&=0 \end{align*}
1.33.1 Solved as second order ode using Kovacic algorithm
Time used: 0.236 (sec)
Writing the ode as
\begin{align*} \left (2 x +1\right ) y^{\prime \prime }-2 y^{\prime }+\left (-2 x -3\right ) y &= 0 \tag {1} \\ A y^{\prime \prime } + B y^{\prime } + C y &= 0 \tag {2} \end{align*}
Comparing (1) and (2) shows that
\begin{align*} A &= 2 x +1 \\ B &= -2\tag {3} \\ C &= -2 x -3 \end{align*}
Applying the Liouville transformation on the dependent variable gives
\begin{align*} z(x) &= y e^{\int \frac {B}{2 A} \,dx} \end{align*}
Then (2) becomes
\begin{align*} z''(x) = r z(x)\tag {4} \end{align*}
Where \(r\) is given by
\begin{align*} r &= \frac {s}{t}\tag {5} \\ &= \frac {2 A B' - 2 B A' + B^2 - 4 A C}{4 A^2} \end{align*}
Substituting the values of \(A,B,C\) from (3) in the above and simplifying gives
\begin{align*} r &= \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\tag {6} \end{align*}
Comparing the above to (5) shows that
\begin{align*} s &= 4 x^{2}+8 x +6\\ t &= \left (2 x +1\right )^{2} \end{align*}
Therefore eq. (4) becomes
\begin{align*} z''(x) &= \left ( \frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\right ) z(x)\tag {7} \end{align*}
Equation (7) is now solved. After finding \(z(x)\) then \(y\) is found using the inverse transformation
\begin{align*} y &= z \left (x \right ) e^{-\int \frac {B}{2 A} \,dx} \end{align*}
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 \(\infty \). The following table
summarizes these cases.
| | |
Case |
Allowed pole order for \(r\) |
Allowed value for \(\mathcal {O}(\infty )\) |
| | |
1 |
\(\left \{ 0,1,2,4,6,8,\cdots \right \} \) |
\(\left \{ \cdots ,-6,-4,-2,0,2,3,4,5,6,\cdots \right \} \) |
| | |
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 |
\(\left \{ 1,2\right \} \) |
\(\left \{ 2,3,4,5,6,7,\cdots \right \} \) |
| | |
Table 33: Necessary conditions for each Kovacic case
The order of \(r\) at \(\infty \) is the degree of \(t\) minus the degree of \(s\). Therefore
\begin{align*} O\left (\infty \right ) &= \text {deg}(t) - \text {deg}(s) \\ &= 2 - 2 \\ &= 0 \end{align*}
The poles of \(r\) in eq. (7) and the order of each pole are determined by solving for the roots
of \(t=\left (2 x +1\right )^{2}\). There is a pole at \(x=-{\frac {1}{2}}\) of order \(2\). Since there is no odd order pole larger than \(2\)
and the order at \(\infty \) is \(0\) 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{align*} L &= [1, 2] \end{align*}
Attempting to find a solution using case \(n=1\).
Looking at poles of order 2. The partial fractions decomposition of \(r\) is
\[
r = 1+\frac {3}{4 \left (x +\frac {1}{2}\right )^{2}}+\frac {1}{x +\frac {1}{2}}
\]
For the pole at \(x=-{\frac {1}{2}}\) let \(b\)
be the coefficient of \(\frac {1}{ \left (x +\frac {1}{2}\right )^{2}}\) in the partial fractions decomposition of \(r\) given above. Therefore \(b={\frac {3}{4}}\). Hence
\begin{alignat*}{2} [\sqrt r]_c &= 0 \\ \alpha _c^{+} &= \frac {1}{2} + \sqrt {1+4 b} &&= {\frac {3}{2}}\\ \alpha _c^{-} &= \frac {1}{2} - \sqrt {1+4 b} &&= -{\frac {1}{2}} \end{alignat*}
Since the order of \(r\) at \(\infty \) is \(O_r(\infty ) = 0\) then
\begin{alignat*}{3} v &= \frac {-O_r(\infty )}{2} &&= \frac {0}{2} &&= 0 \end{alignat*}
\([\sqrt r]_\infty \) is the sum of terms involving \(x^i\) for \(0\leq i \leq v\) in the Laurent series for \(\sqrt r\) at \(\infty \). Therefore
\begin{align*} [\sqrt r]_\infty &= \sum _{i=0}^{v} a_i x^i \\ &= \sum _{i=0}^{0} a_i x^i \tag {8} \end{align*}
Let \(a\) be the coefficient of \(x^v=x^0\) in the above sum. The Laurent series of \(\sqrt r\) at \(\infty \) is
\[ \sqrt r \approx 1+\frac {1}{2 x}-\frac {1}{4 x^{3}}+\frac {11}{32 x^{4}}-\frac {21}{64 x^{5}}+\frac {15}{64 x^{6}}-\frac {3}{32 x^{7}}-\frac {117}{2048 x^{8}} + \dots \tag {9} \]
Comparing Eq. (9)
with Eq. (8) shows that
\[ a = 1 \]
From Eq. (9) the sum up to \(v=0\) gives
\begin{align*} [\sqrt r]_\infty &= \sum _{i=0}^{0} a_i x^i \\ &= 1 \tag {10} \end{align*}
Now we need to find \(b\), where \(b\) be the coefficient of \(x^{v-1} = x^{-1}=\frac {1}{x}\) in \(r\) minus the coefficient of same term but
in \(\left ( [\sqrt r]_\infty \right )^2 \) where \([\sqrt r]_\infty \) was found above in Eq (10). Hence
\[ \left ( [\sqrt r]_\infty \right )^2 = 1 \]
This shows that the coefficient of \(\frac {1}{x}\) in the
above is \(0\). Now we need to find the coefficient of \(\frac {1}{x}\) in \(r\). How this is done depends on if \(v=0\) or not.
Since \(v=0\) then starting from \(r=\frac {s}{t}\) and doing long division in the form
\[ r = Q + \frac {R}{t} \]
Where \(Q\) is the quotient and \(R\) is
the remainder. Then the coefficient of \(\frac {1}{x}\) in \(r\) will be the coefficient in \(R\) of the term in \(x\) of degree
of \(t\) minus one, divided by the leading coefficient in \(t\). Doing long division gives
\begin{align*} r &= \frac {s}{t} \\ &= \frac {4 x^{2}+8 x +6}{4 x^{2}+4 x +1} \\ &= Q + \frac {R}{4 x^{2}+4 x +1}\\ &= \left (1\right ) + \left ( \frac {4 x +5}{4 x^{2}+4 x +1}\right ) \\ &= 1+\frac {4 x +5}{4 x^{2}+4 x +1} \end{align*}
Since the degree of \(t\) is \(2\), then we see that the coefficient of the term \(x\) in the remainder \(R\)
is \(4\). Dividing this by leading coefficient in \(t\) which is \(4\) gives \(1\). Now \(b\) can be found.
\begin{align*} b &= \left (1\right )-\left (0\right )\\ &= 1 \end{align*}
Hence
\begin{alignat*}{3} [\sqrt r]_\infty &= 1\\ \alpha _{\infty }^{+} &= \frac {1}{2} \left ( \frac {b}{a} - v \right ) &&= \frac {1}{2} \left ( \frac {1}{1} - 0 \right ) &&= {\frac {1}{2}}\\ \alpha _{\infty }^{-} &= \frac {1}{2} \left ( -\frac {b}{a} - v \right ) &&= \frac {1}{2} \left ( -\frac {1}{1} - 0 \right ) &&= -{\frac {1}{2}} \end{alignat*}
The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) where \(r\)
is
\[ r=\frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}} \]
| | | | |
pole \(c\) location |
pole order |
\([\sqrt r]_c\) |
\(\alpha _c^{+}\) |
\(\alpha _c^{-}\) |
| | | | |
\(-{\frac {1}{2}}\) | \(2\) | \(0\) | \(\frac {3}{2}\) | \(-{\frac {1}{2}}\) |
| | | | |
| | | |
Order of \(r\) at \(\infty \) |
\([\sqrt r]_\infty \) |
\(\alpha _\infty ^{+}\) |
\(\alpha _\infty ^{-}\) |
| | | |
\(0\) |
\(1\) | \(\frac {1}{2}\) | \(-{\frac {1}{2}}\) |
| | | |
Now that the all \([\sqrt r]_c\) and its associated \(\alpha _c^{\pm }\) have been determined for all the poles in the set \(\Gamma \) and \([\sqrt r]_\infty \)
and its associated \(\alpha _\infty ^{\pm }\) have also been found, the next step is to determine possible non negative
integer \(d\) from these using
\begin{align*} d &= \alpha _\infty ^{s(\infty )} - \sum _{c \in \Gamma } \alpha _c^{s(c)} \end{align*}
Where \(s(c)\) is either \(+\) or \(-\) and \(s(\infty )\) is the sign of \(\alpha _\infty ^{\pm }\). This is done by trial over all set of families \(s=(s(c))_{c \in \Gamma \cup {\infty }}\) until
such \(d\) is found to work in finding candidate \(\omega \). Trying \(\alpha _\infty ^{-} = -{\frac {1}{2}}\) then
\begin{align*} d &= \alpha _\infty ^{-} - \left ( \alpha _{c_1}^{-} \right ) \\ &= -{\frac {1}{2}} - \left ( -{\frac {1}{2}} \right ) \\ &= 0 \end{align*}
Since \(d\) an integer and \(d \geq 0\) then it can be used to find \(\omega \) using
\begin{align*} \omega &= \sum _{c \in \Gamma } \left ( s(c) [\sqrt r]_c + \frac {\alpha _c^{s(c)}}{x-c} \right ) + s(\infty ) [\sqrt r]_\infty \end{align*}
The above gives
\begin{align*} \omega &= \left ( (-)[\sqrt r]_{c_1} + \frac { \alpha _{c_1}^{-} }{x- c_1}\right ) + (-) [\sqrt r]_\infty \\ &= -\frac {1}{2 \left (x +\frac {1}{2}\right )} + (-) \left ( 1 \right ) \\ &= -\frac {1}{2 \left (x +\frac {1}{2}\right )}-1\\ &= -\frac {2 \left (x +1\right )}{2 x +1} \end{align*}
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{align*} p'' + 2 \omega p' + \left ( \omega ' +\omega ^2 -r\right ) p = 0 \tag {1A} \end{align*}
Let
\begin{align*} p(x) &= 1\tag {2A} \end{align*}
Substituting the above in eq. (1A) gives
\begin{align*} \left (0\right ) + 2 \left (-\frac {1}{2 \left (x +\frac {1}{2}\right )}-1\right ) \left (0\right ) + \left ( \left (\frac {1}{2 \left (x +\frac {1}{2}\right )^{2}}\right ) + \left (-\frac {1}{2 \left (x +\frac {1}{2}\right )}-1\right )^2 - \left (\frac {4 x^{2}+8 x +6}{\left (2 x +1\right )^{2}}\right ) \right ) &= 0\\ 0 = 0 \end{align*}
The equation is satisfied since both sides are zero. Therefore the first solution to the ode \(z'' = r z\) is
\begin{align*} z_1(x) &= p e^{ \int \omega \,dx} \\ &= {\mathrm e}^{\int \left (-\frac {1}{2 \left (x +\frac {1}{2}\right )}-1\right )d x}\\ &= \frac {{\mathrm e}^{-x}}{\sqrt {2 x +1}} \end{align*}
The first solution to the original ode in \(y\) is found from
\begin{align*}
y_1 &= z_1 e^{ \int -\frac {1}{2} \frac {B}{A} \,dx} \\
&= z_1 e^{ -\int \frac {1}{2} \frac {-2}{2 x +1} \,dx} \\
&= z_1 e^{\frac {\ln \left (2 x +1\right )}{2}} \\
&= z_1 \left (\sqrt {2 x +1}\right ) \\
\end{align*}
Which simplifies to
\[
y_1 = {\mathrm e}^{-x}
\]
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{align*}
y_2 &= y_1 \int \frac { e^{\int -\frac {-2}{2 x +1} \,dx}}{\left (y_1\right )^2} \,dx \\
&= y_1 \int \frac { e^{\ln \left (2 x +1\right )}}{\left (y_1\right )^2} \,dx \\
&= y_1 \left (x \,{\mathrm e}^{2 x}\right ) \\
\end{align*}
Therefore
the solution is
\begin{align*}
y &= c_1 y_1 + c_2 y_2 \\
&= c_1 \left ({\mathrm e}^{-x}\right ) + c_2 \left ({\mathrm e}^{-x}\left (x \,{\mathrm e}^{2 x}\right )\right ) \\
\end{align*}
Will add steps showing solving for IC soon.
1.33.2 Maple step by step solution
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (2 x +1\right ) \left (\frac {d}{d x}y^{\prime }\right )-2 y^{\prime }-\left (2 x +3\right ) y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d}{d x}y^{\prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\left (2 x +3\right ) y}{2 x +1}+\frac {2 y^{\prime }}{2 x +1} \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE; ODE is linear}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }-\frac {2 y^{\prime }}{2 x +1}-\frac {\left (2 x +3\right ) y}{2 x +1}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=-\frac {1}{2}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=-\frac {2}{2 x +1}, P_{3}\left (x \right )=-\frac {2 x +3}{2 x +1}\right ] \\ {} & \circ & \left (x +\frac {1}{2}\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-\frac {1}{2} \\ {} & {} & \left (\left (x +\frac {1}{2}\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-\frac {1}{2}}}}=-1 \\ {} & \circ & \left (x +\frac {1}{2}\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-\frac {1}{2} \\ {} & {} & \left (\left (x +\frac {1}{2}\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-\frac {1}{2}}}}=0 \\ {} & \circ & x =-\frac {1}{2}\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=-\frac {1}{2}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=-\frac {1}{2} \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (2 x +1\right ) \left (\frac {d}{d x}y^{\prime }\right )-2 y^{\prime }+\left (-2 x -3\right ) y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -\frac {1}{2}\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & 2 u \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )-2 \frac {d}{d u}y \left (u \right )+\left (-2 u -2\right ) y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot y \left (u \right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & u^{m}\cdot y \left (u \right )=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d u}y \left (u \right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d u}y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & \frac {d}{d u}y \left (u \right )=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 2 a_{0} r \left (-2+r \right ) u^{-1+r}+\left (2 a_{1} \left (1+r \right ) \left (-1+r \right )-2 a_{0}\right ) u^{r}+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (2 a_{k +1} \left (k +1+r \right ) \left (k +r -1\right )-2 a_{k}-2 a_{k -1}\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & 2 r \left (-2+r \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, 2\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & 2 a_{1} \left (1+r \right ) \left (-1+r \right )-2 a_{0}=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & 2 a_{k +1} \left (k +1+r \right ) \left (k +r -1\right )-2 a_{k}-2 a_{k -1}=0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & 2 a_{k +2} \left (k +2+r \right ) \left (k +r \right )-2 a_{k +1}-2 a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +2+r \right ) \left (k +r \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +2\right ) k} \\ \bullet & {} & \textrm {Series not valid for}\hspace {3pt} r =0\hspace {3pt}\textrm {, division by}\hspace {3pt} 0\hspace {3pt}\textrm {in the recursion relation at}\hspace {3pt} k =0 \\ {} & {} & a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +2\right ) k} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =2 \\ {} & {} & a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +4\right ) \left (k +2\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =2 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +2}, a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +4\right ) \left (k +2\right )}, 6 a_{1}-2 a_{0}=0\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +\frac {1}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +\frac {1}{2}\right )^{k +2}, a_{k +2}=\frac {a_{k +1}+a_{k}}{\left (k +4\right ) \left (k +2\right )}, 6 a_{1}-2 a_{0}=0\right ] \end {array} \]
1.33.3 Maple trace
Methods for second order ODEs:
1.33.4 Maple dsolve solution
Solving time : 0.004
(sec)
Leaf size : 16
dsolve((2*x+1)*diff(diff(y(x),x),x)-2*diff(y(x),x)-(2*x+3)*y(x) = 0,
y(x),singsol=all)
\[
y = c_1 \,{\mathrm e}^{-x}+c_2 x \,{\mathrm e}^{x}
\]
1.33.5 Mathematica DSolve solution
Solving time : 0.128
(sec)
Leaf size : 29
DSolve[{(2*x+1)*D[y[x],{x,2}]-2*D[y[x],x]-(2*x+3)*y[x]==0,{}},
y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to e^{-x-\frac {1}{2}} \left (c_2 e^{2 x+1} x+c_1\right )
\]