2.2.16 problem 17

Internal problem ID [8815]
Book : Second order enumerated odes
Section : section 2
Problem number : 17
Date solved : Thursday, December 12, 2024 at 09:51:54 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

Solve

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}}&=0 \end{align*}

Solved as second order ode using change of variable on x method 2

Time used: 0.346 (sec)

In normal form the ode

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}}&=0 \tag {1} \end{align*}

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=\frac {2}{x}\\ q \left (x \right )&=\frac {a^{2}}{x^{4}} \end{align*}

Applying change of variables \(\tau = g \left (x \right )\) to (2) gives

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end{align*}

Where \(\tau \) is the new independent variable, and

\begin{align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end{align*}

Let \(p_{1} = 0\). Eq (4) simplifies to

\begin{align*} \tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )&=0 \end{align*}

This ode is solved resulting in

\begin{align*} \tau &= \int {\mathrm e}^{-\int p \left (x \right )d x}d x\\ &= \int {\mathrm e}^{-\int \frac {2}{x}d x}d x\\ &= \int e^{-2 \ln \left (x \right )} \,dx\\ &= \int \frac {1}{x^{2}}d x\\ &= -\frac {1}{x}\tag {6} \end{align*}

Using (6) to evaluate \(q_{1}\) from (5) gives

\begin{align*} q_{1} \left (\tau \right ) &= \frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &= \frac {\frac {a^{2}}{x^{4}}}{\frac {1}{x^{4}}}\\ &= a^{2}\tag {7} \end{align*}

Substituting the above in (3) and noting that now \(p_{1} = 0\) results in

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+q_{1} y \left (\tau \right )&=0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+a^{2} y \left (\tau \right )&=0 \end{align*}

The above ode is now solved for \(y \left (\tau \right )\).This is second order with constant coefficients homogeneous ODE. In standard form the ODE is

\[ A y''(\tau ) + B y'(\tau ) + C y(\tau ) = 0 \]

Where in the above \(A=1, B=0, C=a^{2}\). Let the solution be \(y \left (\tau \right )=e^{\lambda \tau }\). Substituting this into the ODE gives

\[ \lambda ^{2} {\mathrm e}^{\tau \lambda }+a^{2} {\mathrm e}^{\tau \lambda } = 0 \tag {1} \]

Since exponential function is never zero, then dividing Eq(2) throughout by \(e^{\lambda \tau }\) gives

\[ a^{2}+\lambda ^{2} = 0 \tag {2} \]

Equation (2) is the characteristic equation of the ODE. Its roots determine the general solution form.Using the quadratic formula

\[ \lambda _{1,2} = \frac {-B}{2 A} \pm \frac {1}{2 A} \sqrt {B^2 - 4 A C} \]

Substituting \(A=1, B=0, C=a^{2}\) into the above gives

\begin{align*} \lambda _{1,2} &= \frac {0}{(2) \left (1\right )} \pm \frac {1}{(2) \left (1\right )} \sqrt {0^2 - (4) \left (1\right )\left (a^{2}\right )}\\ &= \pm \sqrt {-a^{2}} \end{align*}

Hence

\begin{align*} \lambda _1 &= + \sqrt {-a^{2}} \\ \lambda _2 &= - \sqrt {-a^{2}} \\ \end{align*}

Which simplifies to

\begin{align*} \lambda _1 &= \sqrt {-a^{2}} \\ \lambda _2 &= -\sqrt {-a^{2}} \\ \end{align*}

Since roots are real and distinct, then the solution is

\begin{align*} y \left (\tau \right ) &= c_1 e^{\lambda _1 \tau } + c_2 e^{\lambda _2 \tau } \\ y \left (\tau \right ) &= c_1 e^{\left (\sqrt {-a^{2}}\right )\tau } +c_2 e^{\left (-\sqrt {-a^{2}}\right )\tau } \\ \end{align*}

Or

\[ y \left (\tau \right ) =c_1 \,{\mathrm e}^{\tau \sqrt {-a^{2}}}+c_2 \,{\mathrm e}^{-\tau \sqrt {-a^{2}}} \]

Will add steps showing solving for IC soon.

The above solution is now transformed back to \(y\) using (6) which results in

\[ y = c_1 \,{\mathrm e}^{-\frac {\sqrt {-a^{2}}}{x}}+c_2 \,{\mathrm e}^{\frac {\sqrt {-a^{2}}}{x}} \]

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= c_1 \,{\mathrm e}^{-\frac {\sqrt {-a^{2}}}{x}}+c_2 \,{\mathrm e}^{\frac {\sqrt {-a^{2}}}{x}} \\ \end{align*}

Solved as second order ode using change of variable on x method 1

Time used: 0.091 (sec)

In normal form the ode

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}}&=0 \tag {1} \end{align*}

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=0 \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=\frac {2}{x}\\ q \left (x \right )&=\frac {a^{2}}{x^{4}} \end{align*}

Applying change of variables \(\tau = g \left (x \right )\) to (2) results

\begin{align*} \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+p_{1} \left (\frac {d}{d \tau }y \left (\tau \right )\right )+q_{1} y \left (\tau \right )&=0 \tag {3} \end{align*}

Where \(\tau \) is the new independent variable, and

\begin{align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {4} \\ q_{1} \left (\tau \right ) &=\frac {q \left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\tag {5} \end{align*}

Let \(q_1=c^2\) where \(c\) is some constant. Therefore from (5)

\begin{align*} \tau ' &= \frac {1}{c}\sqrt {q}\\ &= \frac {\sqrt {\frac {a^{2}}{x^{4}}}}{c}\tag {6} \\ \tau '' &= -\frac {2 a^{2}}{c \sqrt {\frac {a^{2}}{x^{4}}}\, x^{5}} \end{align*}

Substituting the above into (4) results in

\begin{align*} p_{1} \left (\tau \right ) &=\frac {\tau ^{\prime \prime }\left (x \right )+p \left (x \right ) \tau ^{\prime }\left (x \right )}{{\tau ^{\prime }\left (x \right )}^{2}}\\ &=\frac {-\frac {2 a^{2}}{c \sqrt {\frac {a^{2}}{x^{4}}}\, x^{5}}+\frac {2}{x}\frac {\sqrt {\frac {a^{2}}{x^{4}}}}{c}}{\left (\frac {\sqrt {\frac {a^{2}}{x^{4}}}}{c}\right )^2} \\ &=0 \end{align*}

Therefore ode (3) now becomes

\begin{align*} y \left (\tau \right )'' + p_1 y \left (\tau \right )' + q_1 y \left (\tau \right ) &= 0 \\ \frac {d^{2}}{d \tau ^{2}}y \left (\tau \right )+c^{2} y \left (\tau \right ) &= 0 \tag {7} \end{align*}

The above ode is now solved for \(y \left (\tau \right )\). Since the ode is now constant coefficients, it can be easily solved to give

\begin{align*} y \left (\tau \right ) &= c_1 \cos \left (c \tau \right )+c_2 \sin \left (c \tau \right ) \end{align*}

Now from (6)

\begin{align*} \tau &= \int \frac {1}{c} \sqrt q \,dx \\ &= \frac {\int \sqrt {\frac {a^{2}}{x^{4}}}d x}{c}\\ &= -\frac {x \sqrt {\frac {a^{2}}{x^{4}}}}{c} \end{align*}

Substituting the above into the solution obtained gives

\[ y = c_1 \cos \left (x \sqrt {\frac {a^{2}}{x^{4}}}\right )-c_2 \sin \left (x \sqrt {\frac {a^{2}}{x^{4}}}\right ) \]

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= c_1 \cos \left (x \sqrt {\frac {a^{2}}{x^{4}}}\right )-c_2 \sin \left (x \sqrt {\frac {a^{2}}{x^{4}}}\right ) \\ \end{align*}

Solved as second order Bessel ode

Time used: 0.070 (sec)

Writing the ode as

\begin{align*} x^{2} y^{\prime \prime }+2 y^{\prime } x +\frac {a^{2} y}{x^{2}} = 0\tag {1} \end{align*}

Bessel ode has the form

\begin{align*} x^{2} y^{\prime \prime }+y^{\prime } x +\left (-n^{2}+x^{2}\right ) y = 0\tag {2} \end{align*}

The generalized form of Bessel ode is given by Bowman (1958) as the following

\begin{align*} x^{2} y^{\prime \prime }+\left (1-2 \alpha \right ) x y^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) y = 0\tag {3} \end{align*}

With the standard solution

\begin{align*} y&=x^{\alpha } \left (c_1 \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_2 \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end{align*}

Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives

\begin{align*} \alpha &= -{\frac {1}{2}}\\ \beta &= a\\ n &= {\frac {1}{2}}\\ \gamma &= -1 \end{align*}

Substituting all the above into (4) gives the solution as

\begin{align*} y = \frac {c_1 \sqrt {2}\, \sin \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}}-\frac {c_2 \sqrt {2}\, \cos \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}} \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= \frac {c_1 \sqrt {2}\, \sin \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}}-\frac {c_2 \sqrt {2}\, \cos \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}} \\ \end{align*}

Solved as second order ode using Kovacic algorithm

Time used: 0.227 (sec)

Writing the ode as

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} &= 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 &= 1 \\ B &= \frac {2}{x}\tag {3} \\ C &= \frac {a^{2}}{x^{4}} \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 {-a^{2}}{x^{4}}\tag {6} \end{align*}

Comparing the above to (5) shows that

\begin{align*} s &= -a^{2}\\ t &= x^{4} \end{align*}

Therefore eq. (4) becomes

\begin{align*} z''(x) &= \left ( -\frac {a^{2}}{x^{4}}\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 2.34: 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) \\ &= 4 - 0 \\ &= 4 \end{align*}

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

\begin{align*} L &= [1] \end{align*}

Attempting to find a solution using case \(n=1\).

Looking at higher order poles of order \(2 v \)≥\( 4\) (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 \(2 \leq i \leq v\) in the Laurent series expansion of \(\sqrt r\) expanded around each pole \(c\). Hence

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

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{alignat*}{1} \alpha _c^{+} &= \frac {1}{2} \left ( \frac {b}{a} + v \right ) \\ \alpha _c^{-} &= \frac {1}{2} \left (- \frac {b}{a} + v \right ) \end{alignat*}

The partial fraction decomposition of \(r\) is

\[ r = -\frac {a^{2}}{x^{4}} \]

There is pole in \(r\) at \(x= 0\) of order \(4\), hence \(v=2\). Expanding \(\sqrt {r}\) as Laurent series about this pole \(c=0\) gives

\[ [\sqrt {r}]_c \approx \frac {i a}{x^{2}} + \dots \tag {2B} \]

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

\[ [\sqrt {r}]_c = \frac {i a}{x^{2}} \tag {3B} \]

The above shows that the coefficient of \(\frac {1}{(x-0)^{2}}\) is

\[ a = i a \]

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 \(c=0\). This term becomes \(\frac {1}{x^{3}}\). 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 \(0\). Therefore

\begin{align*} b &= \left (0\right )-(0)\\ &= 0 \end{align*}

Hence

\begin{alignat*}{3} [\sqrt r]_c &= \frac {i a}{x^{2}} \\ \alpha _c^{+} &= \frac {1}{2} \left ( \frac {b}{a} + v \right ) &&= \frac {1}{2} \left ( \frac {0}{i a} + 2 \right ) &&=1\\ \alpha _c^{-} &= \frac {1}{2} \left (- \frac {b}{a} + v \right ) &&= \frac {1}{2} \left (- \frac {0}{i a} + 2 \right )&&=1 \end{alignat*}

Since the order of \(r\) at \(\infty \) is \(4 > 2\) then

\begin{alignat*}{2} [\sqrt r]_\infty &= 0 \\ \alpha _{\infty }^{+} &= 0 \\ \alpha _{\infty }^{-} &= 1 \end{alignat*}

The following table summarizes the findings so far for poles and for the order of \(r\) at \(\infty \) where \(r\) is

\[ r=-\frac {a^{2}}{x^{4}} \]

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

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

Now that the all \([\sqrt r]_c\) and its associated \(\alpha _c^{\pm }\) have been determined for all the poles in the set \(\Gamma \) 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 \(\alpha _\infty ^{-} = 1\) then

\begin{align*} d &= \alpha _\infty ^{-} - \left ( \alpha _{c_1}^{-} \right ) \\ &= 1 - \left ( 1 \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 {i a}{x^{2}}+\frac {1}{x} + (-) \left ( 0 \right ) \\ &= -\frac {i a}{x^{2}}+\frac {1}{x}\\ &= \frac {-i a +x}{x^{2}} \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 {i a}{x^{2}}+\frac {1}{x}\right ) \left (0\right ) + \left ( \left (\frac {2 i a}{x^{3}}-\frac {1}{x^{2}}\right ) + \left (-\frac {i a}{x^{2}}+\frac {1}{x}\right )^2 - \left (-\frac {a^{2}}{x^{4}}\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 {i a}{x^{2}}+\frac {1}{x}\right )d x}\\ &= x \,{\mathrm e}^{\frac {i a}{x}} \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 {\frac {2}{x}}{1} \,dx} \\ &= z_1 e^{-\ln \left (x \right )} \\ &= z_1 \left (\frac {1}{x}\right ) \\ \end{align*}

Which simplifies to

\[ y_1 = {\mathrm e}^{\frac {i a}{x}} \]

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 {\frac {2}{x}}{1} \,dx}}{\left (y_1\right )^2} \,dx \\ &= y_1 \int \frac { e^{-2 \ln \left (x \right )}}{\left (y_1\right )^2} \,dx \\ &= y_1 \left (-\frac {i {\mathrm e}^{-\frac {2 i a}{x}}}{2 a}\right ) \\ \end{align*}

Therefore the solution is

\begin{align*} y &= c_1 y_1 + c_2 y_2 \\ &= c_1 \left ({\mathrm e}^{\frac {i a}{x}}\right ) + c_2 \left ({\mathrm e}^{\frac {i a}{x}}\left (-\frac {i {\mathrm e}^{-\frac {2 i a}{x}}}{2 a}\right )\right ) \\ \end{align*}

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= c_1 \,{\mathrm e}^{\frac {i a}{x}}-\frac {i c_2 \,{\mathrm e}^{-\frac {i a}{x}}}{2 a} \\ \end{align*}

Solved as second order ode adjoint method

Time used: 0.441 (sec)

In normal form the ode

\begin{align*} y^{\prime \prime }+\frac {2 y^{\prime }}{x}+\frac {a^{2} y}{x^{4}} = 0 \tag {1} \end{align*}

Becomes

\begin{align*} y^{\prime \prime }+p \left (x \right ) y^{\prime }+q \left (x \right ) y&=r \left (x \right ) \tag {2} \end{align*}

Where

\begin{align*} p \left (x \right )&=\frac {2}{x}\\ q \left (x \right )&=\frac {a^{2}}{x^{4}}\\ r \left (x \right )&=0 \end{align*}

The Lagrange adjoint ode is given by

\begin{align*} \xi ^{''}-(\xi \, p)'+\xi q &= 0\\ \xi ^{''}-\left (\frac {2 \xi \left (x \right )}{x}\right )' + \left (\frac {a^{2} \xi \left (x \right )}{x^{4}}\right ) &= 0\\ \xi ^{\prime \prime }\left (x \right )-\frac {2 \xi ^{\prime }\left (x \right )}{x}+\frac {\left (a^{2}+2 x^{2}\right ) \xi \left (x \right )}{x^{4}}&= 0 \end{align*}

Which is solved for \(\xi (x)\). Writing the ode as

\begin{align*} x^{2} \xi ^{\prime \prime }-2 \xi ^{\prime } x +\left (2+\frac {a^{2}}{x^{2}}\right ) \xi = 0\tag {1} \end{align*}

Bessel ode has the form

\begin{align*} x^{2} \xi ^{\prime \prime }+\xi ^{\prime } x +\left (-n^{2}+x^{2}\right ) \xi = 0\tag {2} \end{align*}

The generalized form of Bessel ode is given by Bowman (1958) as the following

\begin{align*} x^{2} \xi ^{\prime \prime }+\left (1-2 \alpha \right ) x \xi ^{\prime }+\left (\beta ^{2} \gamma ^{2} x^{2 \gamma }-n^{2} \gamma ^{2}+\alpha ^{2}\right ) \xi = 0\tag {3} \end{align*}

With the standard solution

\begin{align*} \xi &=x^{\alpha } \left (c_1 \operatorname {BesselJ}\left (n , \beta \,x^{\gamma }\right )+c_2 \operatorname {BesselY}\left (n , \beta \,x^{\gamma }\right )\right )\tag {4} \end{align*}

Comparing (3) to (1) and solving for \(\alpha ,\beta ,n,\gamma \) gives

\begin{align*} \alpha &= {\frac {3}{2}}\\ \beta &= a\\ n &= -{\frac {1}{2}}\\ \gamma &= -1 \end{align*}

Substituting all the above into (4) gives the solution as

\begin{align*} \xi = \frac {c_1 \,x^{{3}/{2}} \sqrt {2}\, \cos \left (\frac {a}{x}\right )}{\sqrt {\pi }\, \sqrt {\frac {a}{x}}}+\frac {c_2 \,x^{{3}/{2}} \sqrt {2}\, \sin \left (\frac {a}{x}\right )}{\sqrt {\pi }\, \sqrt {\frac {a}{x}}} \end{align*}

Will add steps showing solving for IC soon.

The original ode now reduces to first order ode

\begin{align*} \xi \left (x \right ) y^{\prime }-y \xi ^{\prime }\left (x \right )+\xi \left (x \right ) p \left (x \right ) y&=\int \xi \left (x \right ) r \left (x \right )d x\\ y^{\prime }+y \left (p \left (x \right )-\frac {\xi ^{\prime }\left (x \right )}{\xi \left (x \right )}\right )&=\frac {\int \xi \left (x \right ) r \left (x \right )d x}{\xi \left (x \right )} \end{align*}

Or

\begin{align*} y^{\prime }+y \left (\frac {2}{x}-\frac {\frac {3 c_1 \sqrt {x}\, \sqrt {2}\, \cos \left (\frac {a}{x}\right )}{2 \sqrt {\pi }\, \sqrt {\frac {a}{x}}}+\frac {c_1 \sqrt {2}\, \cos \left (\frac {a}{x}\right ) a}{2 \sqrt {x}\, \sqrt {\pi }\, \left (\frac {a}{x}\right )^{{3}/{2}}}+\frac {c_1 \sqrt {2}\, a \sin \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}}+\frac {3 c_2 \sqrt {x}\, \sqrt {2}\, \sin \left (\frac {a}{x}\right )}{2 \sqrt {\pi }\, \sqrt {\frac {a}{x}}}+\frac {c_2 \sqrt {2}\, \sin \left (\frac {a}{x}\right ) a}{2 \sqrt {x}\, \sqrt {\pi }\, \left (\frac {a}{x}\right )^{{3}/{2}}}-\frac {c_2 \sqrt {2}\, a \cos \left (\frac {a}{x}\right )}{\sqrt {x}\, \sqrt {\pi }\, \sqrt {\frac {a}{x}}}}{\frac {c_1 \,x^{{3}/{2}} \sqrt {2}\, \cos \left (\frac {a}{x}\right )}{\sqrt {\pi }\, \sqrt {\frac {a}{x}}}+\frac {c_2 \,x^{{3}/{2}} \sqrt {2}\, \sin \left (\frac {a}{x}\right )}{\sqrt {\pi }\, \sqrt {\frac {a}{x}}}}\right )&=0 \end{align*}

Which is now a first order ode. This is now solved for \(y\). In canonical form a linear first order is

\begin{align*} y^{\prime } + q(x)y &= p(x) \end{align*}

Comparing the above to the given ode shows that

\begin{align*} q(x) &=-\frac {\left (-\cos \left (\frac {a}{x}\right ) c_2 +\sin \left (\frac {a}{x}\right ) c_1 \right ) a}{\left (c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )\right ) x^{2}}\\ p(x) &=0 \end{align*}

The integrating factor \(\mu \) is

\begin{align*} \mu &= e^{\int {q\,dx}}\\ &= {\mathrm e}^{\int -\frac {\left (-\cos \left (\frac {a}{x}\right ) c_2 +\sin \left (\frac {a}{x}\right ) c_1 \right ) a}{\left (c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )\right ) x^{2}}d x}\\ &= \frac {1}{c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )} \end{align*}

The ode becomes

\begin{align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \mu y &= 0 \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\frac {y}{c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )}\right ) &= 0 \end{align*}

Integrating gives

\begin{align*} \frac {y}{c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )}&= \int {0 \,dx} + c_3 \\ &=c_3 \end{align*}

Dividing throughout by the integrating factor \(\frac {1}{c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )}\) gives the final solution

\[ y = \left (c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )\right ) c_3 \]

Hence, the solution found using Lagrange adjoint equation method is

\begin{align*} y &= \left (c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right )\right ) c_3 \\ \end{align*}

The constants can be merged to give

\[ y = c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right ) \]

Will add steps showing solving for IC soon.

Summary of solutions found

\begin{align*} y &= c_1 \cos \left (\frac {a}{x}\right )+c_2 \sin \left (\frac {a}{x}\right ) \\ \end{align*}

Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )+\frac {2 \left (\frac {d}{d x}y \left (x \right )\right )}{x}+\frac {a^{2} y \left (x \right )}{x^{4}}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right ) \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {d^{2}}{d x^{2}}y \left (x \right )\right ) x^{4}+2 \left (\frac {d}{d x}y \left (x \right )\right ) x^{3}+a^{2} y \left (x \right )=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =\ln \left (x \right ) \\ \square & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {1st}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\left (\frac {d}{d t}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {d}{d t}y \left (t \right )}{x} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {2nd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right ) \left (\frac {d}{d x}t \left (x \right )\right )^{2}+\left (\frac {d^{2}}{d x^{2}}t \left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d x^{2}}y \left (x \right )=\frac {\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {\frac {d^{2}}{d t^{2}}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right ) x^{4}+2 \left (\frac {d}{d t}y \left (t \right )\right ) x^{2}+a^{2} y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+\left (\frac {d}{d t}y \left (t \right )\right ) x^{2}+a^{2} y \left (t \right )=0 \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d^{2}}{d t^{2}}y \left (t \right )=-\frac {a^{2} y \left (t \right )}{x^{2}}-\frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (t \right )\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^{2}}{d t^{2}}y \left (t \right )+\frac {d}{d t}y \left (t \right )+\frac {a^{2} y \left (t \right )}{x^{2}}=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}+r +\frac {a^{2}}{x^{2}}=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \frac {r^{2} x^{2}+r \,x^{2}+a^{2}}{x^{2}}=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\frac {-\frac {x}{2}+\frac {\sqrt {-4 a^{2}+x^{2}}}{2}}{x}, \frac {-\frac {x}{2}-\frac {\sqrt {-4 a^{2}+x^{2}}}{2}}{x}\right ) \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{\frac {\left (-\frac {x}{2}+\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {2nd solution of the ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{\frac {\left (-\frac {x}{2}-\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=\mathit {C1} y_{1}\left (t \right )+\mathit {C2} y_{2}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y \left (t \right )=\mathit {C1} \,{\mathrm e}^{\frac {\left (-\frac {x}{2}+\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) t}{x}}+\mathit {C2} \,{\mathrm e}^{\frac {\left (-\frac {x}{2}-\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y \left (x \right )=\mathit {C1} \,{\mathrm e}^{\frac {\left (-\frac {x}{2}+\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) \ln \left (x \right )}{x}}+\mathit {C2} \,{\mathrm e}^{\frac {\left (-\frac {x}{2}-\frac {\sqrt {-4 a^{2}+x^{2}}}{2}\right ) \ln \left (x \right )}{x}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y \left (x \right )=\mathit {C1} \,x^{\frac {-x +\sqrt {-4 a^{2}+x^{2}}}{2 x}}+\mathit {C2} \,x^{-\frac {x +\sqrt {-4 a^{2}+x^{2}}}{2 x}} \end {array} \]

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)] 
<- linear_1 successful`
 

Maple dsolve solution

Solving time : 0.004 (sec)
Leaf size : 21

dsolve(diff(diff(y(x),x),x)+2/x*diff(y(x),x)+a^2/x^4*y(x) = 0, 
       y(x),singsol=all)
 

\[ y = c_{1} \sin \left (\frac {a}{x}\right )+c_{2} \cos \left (\frac {a}{x}\right ) \]

Mathematica DSolve solution

Solving time : 0.031 (sec)
Leaf size : 25

DSolve[{D[y[x],{x,2}]+2/x*D[y[x],x]+a^2/x^4*y[x]==0,{}}, 
       y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_1 \cos \left (\frac {a}{x}\right )-c_2 \sin \left (\frac {a}{x}\right ) \]