Internal problem ID [12178]
Internal file name [OUTPUT/10831_Thursday_September_21_2023_05_47_34_AM_92910118/index.tex]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 2, DIFFERENTIAL EQUATIONS OF THE SECOND ORDER AND HIGHER.
Problems page 172
Problem number: Problem 16.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "second_order_bessel_ode"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (9 x^{2}-\frac {1}{25}\right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (9 x^{2}-\frac {1}{25}\right ) y = 0\tag {1} \end {align*}
Bessel ode has the form \begin {align*} x^{2} y^{\prime \prime }+x y^{\prime }+\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 &= 0\\ \beta &= 3\\ n &= -{\frac {1}{5}}\\ \gamma &= 1 \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} \operatorname {BesselJ}\left (-\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (-\frac {1}{5}, 3 x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \operatorname {BesselJ}\left (-\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (-\frac {1}{5}, 3 x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \operatorname {BesselJ}\left (-\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (-\frac {1}{5}, 3 x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x y^{\prime }+\left (9 x^{2}-\frac {1}{25}\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 (225 x^{2}-1\right ) y}{25 x^{2}}-\frac {y^{\prime }}{x} \\ \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 {y^{\prime }}{x}+\frac {\left (225 x^{2}-1\right ) y}{25 x^{2}}=0 \\ \bullet & {} & \textrm {Simplify ODE}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x y^{\prime }+9 y x^{2}-\frac {y}{25}=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =3 x \\ \bullet & {} & \textrm {Compute}\hspace {3pt} y^{\prime } \\ {} & {} & y^{\prime }=3 \frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Compute second derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=9 \frac {d}{d t}\frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Apply change of variables to the ODE}\hspace {3pt} \\ {} & {} & t^{2} \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right )+t \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) t^{2}-\frac {y \left (t \right )}{25}=0 \\ \bullet & {} & \textrm {ODE is now of the Bessel form}\hspace {3pt} \\ \bullet & {} & \textrm {Solution to Bessel ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=c_{1} \mathit {BesselJ}\left (\frac {1}{5}, t\right )+c_{2} \mathit {BesselY}\left (\frac {1}{5}, t\right ) \\ \bullet & {} & \textrm {Make the change from}\hspace {3pt} t \hspace {3pt}\textrm {back to}\hspace {3pt} x \\ {} & {} & y=c_{1} \mathit {BesselJ}\left (\frac {1}{5}, 3 x \right )+c_{2} \mathit {BesselY}\left (\frac {1}{5}, 3 x \right ) \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)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm <- No Liouvillian solutions exists -> Trying a solution in terms of special functions: -> Bessel <- Bessel successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 19
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(9*x^2-1/25)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {BesselJ}\left (\frac {1}{5}, 3 x \right )+c_{2} \operatorname {BesselY}\left (\frac {1}{5}, 3 x \right ) \]
✓ Solution by Mathematica
Time used: 0.031 (sec). Leaf size: 26
DSolve[x^2*y''[x]+x*y'[x]+(9*x^2-1/25)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \operatorname {BesselJ}\left (\frac {1}{5},3 x\right )+c_2 \operatorname {BesselY}\left (\frac {1}{5},3 x\right ) \]