Internal problem ID [9623]
Internal file name [OUTPUT/8564_Monday_June_06_2022_03_52_37_AM_11739899/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1295.
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]]
\[ \boxed {a \,x^{2} y^{\prime \prime }+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) y=0} \]
Writing the ode as \begin {align*} x^{2} y^{\prime \prime }+\frac {b x y^{\prime }}{a}+\left (\frac {c \,x^{2}}{a}+\frac {d x}{a}+\frac {f}{a}\right ) y = 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 &= 2\\ n &= \frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{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 {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\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 {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselJ}\left (\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right )+c_{2} x^{\frac {1}{2}-\frac {b}{2 a}} \operatorname {BesselY}\left (\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{a}, 2 \sqrt {x}\right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a \,x^{2} \left (\frac {d}{d x}y^{\prime }\right )+b x y^{\prime }+\left (c \,x^{2}+d x +f \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 (c \,x^{2}+d x +f \right ) y}{a \,x^{2}}-\frac {b y^{\prime }}{x a} \\ \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 }}{x a}+\frac {\left (c \,x^{2}+d x +f \right ) y}{a \,x^{2}}=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}{x a}, P_{3}\left (x \right )=\frac {c \,x^{2}+d x +f}{a \,x^{2}}\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}}}=\frac {f}{a} \\ {} & \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^{2} \left (\frac {d}{d x}y^{\prime }\right )+b x y^{\prime }+\left (c \,x^{2}+d x +f \right ) 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} x^{m}\cdot y\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..2 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +r +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =m}{\sum }}a_{k -m} x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x \cdot y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x \cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{2}\cdot \left (\frac {d}{d x}y^{\prime }\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & x^{2}\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} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & a_{0} \left (a \,r^{2}-a r +b r +f \right ) x^{r}+\left (\left (a \,r^{2}+a r +b r +b +f \right ) a_{1}+a_{0} d \right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (a \,k^{2}+2 a k r +a \,r^{2}-a k -a r +b k +b r +f \right )+a_{k -1} d +a_{k -2} c \right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & a \,r^{2}-a r +b r +f =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}, -\frac {-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (a \,r^{2}+a r +b r +b +f \right ) a_{1}+a_{0} d =0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {a_{0} d}{a \,r^{2}+a r +b r +b +f} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (\left (k +r \right ) \left (k +r -1\right ) a +b k +b r +f \right ) a_{k}+a_{k -1} d +a_{k -2} c =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2+r \right ) \left (k +1+r \right ) a +b \left (k +2\right )+b r +f \right ) a_{k +2}+a_{k +1} d +a_{k} c =0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {a_{k} c +a_{k +1} d}{a \,k^{2}+2 a k r +a \,r^{2}+3 a k +3 a r +b k +b r +2 a +2 b +f} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a} \\ {} & {} & a_{k +2}=-\frac {a_{k} c +a_{k +1} d}{a \,k^{2}+k \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}+\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k +\frac {b \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}}, a_{k +2}=-\frac {a_{k} c +a_{k +1} d}{a \,k^{2}+k \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}+\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k +\frac {b \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}, a_{1}=-\frac {a_{0} d}{\frac {\left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+\frac {a}{2}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+\frac {b \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a} \\ {} & {} & a_{k +2}=-\frac {a_{k} c +a_{k +1} d}{a \,k^{2}-k \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}-\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k -\frac {b \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}}, a_{k +2}=-\frac {a_{k} c +a_{k +1} d}{a \,k^{2}-k \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}-\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k -\frac {b \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}, a_{1}=-\frac {a_{0} d}{\frac {\left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+\frac {a}{2}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}-\frac {b \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}e_{k} x^{k +\frac {a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}g_{k} x^{k -\frac {-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2 a}}\right ), e_{k +2}=-\frac {c e_{k}+d e_{k +1}}{a \,k^{2}+k \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}+\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k +\frac {b \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}, e_{1}=-\frac {e_{0} d}{\frac {\left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+\frac {a}{2}+\frac {b}{2}+\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+\frac {b \left (a -b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}, g_{k +2}=-\frac {c g_{k}+d g_{k +1}}{a \,k^{2}-k \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )+\frac {\left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+3 a k +\frac {7 a}{2}+\frac {b}{2}-\frac {3 \sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}+b k -\frac {b \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}, g_{1}=-\frac {g_{0} d}{\frac {\left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )^{2}}{4 a}+\frac {a}{2}+\frac {b}{2}-\frac {\sqrt {a^{2}-2 b a -4 f a +b^{2}}}{2}-\frac {b \left (-a +b +\sqrt {a^{2}-2 b a -4 f a +b^{2}}\right )}{2 a}+f}\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 -> elliptic -> Legendre -> Whittaker -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Whittaker successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 106
dsolve(a*x^2*diff(diff(y(x),x),x)+b*x*diff(y(x),x)+(c*x^2+d*x+f)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {b}{2 a}} \left (\operatorname {WhittakerM}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i d}{2 \sqrt {a}\, \sqrt {c}}, \frac {\sqrt {a^{2}+\left (-2 b -4 f \right ) a +b^{2}}}{2 a}, \frac {2 i \sqrt {c}\, x}{\sqrt {a}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.19 (sec). Leaf size: 229
DSolve[(f + d*x + c*x^2)*y[x] + b*x*y'[x] + a*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {i \sqrt {c} x}{\sqrt {a}}} x^{\frac {\sqrt {a^2-2 a (b+2 f)+b^2}+a-b}{2 a}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {a+\frac {i d \sqrt {a}}{\sqrt {c}}+\sqrt {a^2-2 (b+2 f) a+b^2}}{2 a},\frac {a+\sqrt {a^2-2 (b+2 f) a+b^2}}{a},\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )+c_2 L_{-\frac {a+\frac {i d \sqrt {a}}{\sqrt {c}}+\sqrt {a^2-2 (b+2 f) a+b^2}}{2 a}}^{\frac {\sqrt {a^2-2 (b+2 f) a+b^2}}{a}}\left (\frac {2 i \sqrt {c} x}{\sqrt {a}}\right )\right ) \]