Internal problem ID [15492]
Internal file name [OUTPUT/15493_Friday_May_10_2024_05_47_28_PM_91538688/index.tex]
Book: A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV,
G.I. MARKARENKO. MIR, MOSCOW. 1983
Section: Chapter 2 (Higher order ODE’s). Section 18.2. Expanding a solution in generalized power
series. Bessels equation. Exercises page 177
Problem number: 751.
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]]
\[ \boxed {y^{\prime \prime }+\frac {3 y^{\prime }}{x}+4 y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+4 y x^{2} = 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 &= -1\\ \beta &= 2\\ n &= 1\\ \gamma &= 1 \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = \frac {c_{1} \operatorname {BesselJ}\left (1, 2 x \right )}{x}+\frac {c_{2} \operatorname {BesselY}\left (1, 2 x \right )}{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {c_{1} \operatorname {BesselJ}\left (1, 2 x \right )}{x}+\frac {c_{2} \operatorname {BesselY}\left (1, 2 x \right )}{x} \\ \end{align*}
Verification of solutions
\[ y = \frac {c_{1} \operatorname {BesselJ}\left (1, 2 x \right )}{x}+\frac {c_{2} \operatorname {BesselY}\left (1, 2 x \right )}{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime } x +4 y x +3 y^{\prime }=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=-\frac {3 y^{\prime }}{x}-4 y \\ \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} \\ {} & {} & y^{\prime \prime }+\frac {3 y^{\prime }}{x}+4 y=0 \\ \bullet & {} & \textrm {Simplify ODE}\hspace {3pt} \\ {} & {} & x^{2} y^{\prime \prime }+3 x y^{\prime }+4 y x^{2}=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =2 x \\ \bullet & {} & \textrm {Compute}\hspace {3pt} y^{\prime } \\ {} & {} & y^{\prime }=2 \frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Compute second derivative}\hspace {3pt} \\ {} & {} & y^{\prime \prime }=4 \frac {d^{2}}{d t^{2}}y \left (t \right ) \\ \bullet & {} & \textrm {Apply change of variables to the ODE}\hspace {3pt} \\ {} & {} & t^{2} \left (\frac {d^{2}}{d t^{2}}y \left (t \right )\right )+3 t \left (\frac {d}{d t}y \left (t \right )\right )+y \left (t \right ) t^{2}=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & y \left (t \right )=\frac {u \left (t \right )}{t} \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d}{d t}y \left (t \right ) \\ {} & {} & \frac {d}{d t}y \left (t \right )=-\frac {u \left (t \right )}{t^{2}}+\frac {\frac {d}{d t}u \left (t \right )}{t} \\ \bullet & {} & \textrm {Compute}\hspace {3pt} \frac {d^{2}}{d t^{2}}y \left (t \right ) \\ {} & {} & \frac {d^{2}}{d t^{2}}y \left (t \right )=\frac {2 u \left (t \right )}{t^{3}}-\frac {2 \left (\frac {d}{d t}u \left (t \right )\right )}{t^{2}}+\frac {\frac {d^{2}}{d t^{2}}u \left (t \right )}{t} \\ \bullet & {} & \textrm {Apply change of variables to the ODE}\hspace {3pt} \\ {} & {} & t^{2} u \left (t \right )+\left (\frac {d^{2}}{d t^{2}}u \left (t \right )\right ) t^{2}+\left (\frac {d}{d t}u \left (t \right )\right ) t -u \left (t \right )=0 \\ \bullet & {} & \textrm {ODE is now of the Bessel form}\hspace {3pt} \\ \bullet & {} & \textrm {Solution to Bessel ODE}\hspace {3pt} \\ {} & {} & u \left (t \right )=c_{1} \mathit {BesselJ}\left (1, t\right )+c_{2} \mathit {BesselY}\left (1, t\right ) \\ \bullet & {} & \textrm {Make the change from}\hspace {3pt} y\hspace {3pt}\textrm {back to}\hspace {3pt} y \left (t \right ) \\ {} & {} & y \left (t \right )=\frac {c_{1} \mathit {BesselJ}\left (1, t\right )+c_{2} \mathit {BesselY}\left (1, t\right )}{t} \\ \bullet & {} & \textrm {Make the change from}\hspace {3pt} t \hspace {3pt}\textrm {back to}\hspace {3pt} x \\ {} & {} & y=\frac {c_{1} \mathit {BesselJ}\left (1, 2 x \right )+c_{2} \mathit {BesselY}\left (1, 2 x \right )}{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)] 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: 23
dsolve(diff(y(x),x$2)+3/x*diff(y(x),x)+4*y(x)=0,y(x), singsol=all)
\[ y = \frac {c_{1} \operatorname {BesselJ}\left (1, 2 x \right )+c_{2} \operatorname {BesselY}\left (1, 2 x \right )}{x} \]
✓ Solution by Mathematica
Time used: 0.017 (sec). Leaf size: 26
DSolve[y''[x]+3/x*y'[x]+4*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 \operatorname {BesselJ}(1,2 x)+c_2 \operatorname {BesselY}(1,2 x)}{x} \]