Internal problem ID [9469]
Internal file name [OUTPUT/8409_Monday_June_06_2022_02_58_49_AM_16851092/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1140.
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
[[_Emden, _Fowler]]
\[ \boxed {a x y^{\prime \prime }+b y^{\prime }+c y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\frac {x b y^{\prime }}{a}+\frac {c x y}{a} = 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}-\frac {b}{2 a}\\ \beta &= \frac {2 \sqrt {c a}}{a}\\ n &= \frac {-b +a}{a}\\ \gamma &= {\frac {1}{2}} \end {align*}
Substituting all the above into (4) gives the solution as \begin {align*} y = c_{1} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselJ}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselJ}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselJ}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {-b +a}{a}, \frac {2 \sqrt {c a}\, \sqrt {x}}{a}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a x \left (\frac {d}{d x}y^{\prime }\right )+b y^{\prime }+c 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 {b y^{\prime }}{a x}-\frac {c y}{a 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 {b y^{\prime }}{a x}+\frac {c y}{a x}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {b}{a x}, P_{3}\left (x \right )=\frac {c}{a x}\right ] \\ {} & \circ & x \cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x \cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=\frac {b}{a} \\ {} & \circ & x^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =0 \\ {} & {} & \left (x^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}0}}}=0 \\ {} & \circ & x =0\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=0\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & a x \left (\frac {d}{d x}y^{\prime }\right )+b y^{\prime }+c y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & y^{\prime }=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) x^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & x \cdot \left (\frac {d}{d x}y^{\prime }\right )=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) x^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} r \left (a r -a +b \right ) x^{r -1}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (a_{k +1} \left (k +1+r \right ) \left (a \left (k +1\right )+a r -a +b \right )+c a_{k}\right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & r \left (a r -a +b \right )=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {-b +a}{a}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k +1+r \right ) \left (\left (k +r \right ) a +b \right ) a_{k +1}+c a_{k}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=-\frac {c a_{k}}{\left (k +1+r \right ) \left (a k +a r +b \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=-\frac {c a_{k}}{\left (k +1\right ) \left (a k +b \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +1}=-\frac {c a_{k}}{\left (k +1\right ) \left (a k +b \right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {-b +a}{a} \\ {} & {} & a_{k +1}=-\frac {c a_{k}}{\left (k +1+\frac {-b +a}{a}\right ) \left (a k +a \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {-b +a}{a} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {-b +a}{a}}, a_{k +1}=-\frac {c a_{k}}{\left (k +1+\frac {-b +a}{a}\right ) \left (a k +a \right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k +\frac {-b +a}{a}}\right ), d_{k +1}=-\frac {c d_{k}}{\left (k +1\right ) \left (a k +b \right )}, e_{k +1}=-\frac {c e_{k}}{\left (k +1+\frac {-b +a}{a}\right ) \left (a k +a \right )}\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: 134
dsolve(a*x*diff(diff(y(x),x),x)+b*diff(y(x),x)+c*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {x^{-\frac {b}{2 a}} \left (-\sqrt {x}\, \operatorname {BesselJ}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{1} a -\sqrt {x}\, \operatorname {BesselY}\left (\frac {a +b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) \sqrt {\frac {c}{a}}\, c_{2} a +\operatorname {BesselJ}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{1} b +\operatorname {BesselY}\left (\frac {b}{a}, 2 \sqrt {\frac {c}{a}}\, \sqrt {x}\right ) c_{2} b \right )}{a \sqrt {\frac {c}{a}}} \]
✓ Solution by Mathematica
Time used: 0.105 (sec). Leaf size: 120
DSolve[c*y[x] + b*y'[x] + a*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to a^{\frac {1}{2} \left (\frac {b}{a}-1\right )} c^{\frac {a-b}{2 a}} x^{\frac {a-b}{2 a}} \left (c_1 \operatorname {Gamma}\left (\frac {b}{a}\right ) \operatorname {BesselJ}\left (\frac {b}{a}-1,\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )+c_2 \operatorname {Gamma}\left (2-\frac {b}{a}\right ) \operatorname {BesselJ}\left (1-\frac {b}{a},\frac {2 \sqrt {c} \sqrt {x}}{\sqrt {a}}\right )\right ) \]