Internal problem ID [10929]
Internal file name [OUTPUT/10186_Sunday_December_31_2023_11_03_44_AM_97751028/index.tex]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-3 Equation of form
\((a x + b)y''+f(x)y'+g(x)y=0\)
Problem number: 106.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
Unable to solve or complete the solution.
\[ \boxed {\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (a_{1} x +a_{0} \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} 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_{1} m y}{a_{1} x +a_{0}}-\frac {\left (b_{1} x +b_{0} \right ) y^{\prime }}{a_{1} x +a_{0}} \\ \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 {\left (b_{1} x +b_{0} \right ) y^{\prime }}{a_{1} x +a_{0}}-\frac {b_{1} m y}{a_{1} x +a_{0}}=0 \\ \square & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=-\frac {a_{0}}{a_{1}}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & \circ & \textrm {Define functions}\hspace {3pt} \\ {} & {} & \left [P_{2}\left (x \right )=\frac {b_{1} x +b_{0}}{a_{1} x +a_{0}}, P_{3}\left (x \right )=-\frac {b_{1} m}{a_{1} x +a_{0}}\right ] \\ {} & \circ & \left (x +\frac {a_{0}}{a_{1}}\right )\cdot P_{2}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-\frac {a_{0}}{a_{1}} \\ {} & {} & \left (\left (x +\frac {a_{0}}{a_{1}}\right )\cdot P_{2}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-\frac {a_{0}}{a_{1}}}}}=\frac {-\frac {b_{1} a_{0}}{a_{1}}+b_{0}}{a_{1}} \\ {} & \circ & \left (x +\frac {a_{0}}{a_{1}}\right )^{2}\cdot P_{3}\left (x \right )\textrm {is analytic at}\hspace {3pt} x =-\frac {a_{0}}{a_{1}} \\ {} & {} & \left (\left (x +\frac {a_{0}}{a_{1}}\right )^{2}\cdot P_{3}\left (x \right )\right )\bigg | {\mstack {}{_{x \hiderel {=}-\frac {a_{0}}{a_{1}}}}}=0 \\ {} & \circ & x =-\frac {a_{0}}{a_{1}}\textrm {is a regular singular point}\hspace {3pt} \\ & {} & \textrm {Check to see if}\hspace {3pt} x_{0}=-\frac {a_{0}}{a_{1}}\hspace {3pt}\textrm {is a regular singular point}\hspace {3pt} \\ {} & {} & x_{0}=-\frac {a_{0}}{a_{1}} \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & \left (a_{1} x +a_{0} \right ) \left (\frac {d}{d x}y^{\prime }\right )+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y=0 \\ \bullet & {} & \textrm {Change variables using}\hspace {3pt} x =u -\frac {a_{0}}{a_{1}}\hspace {3pt}\textrm {so that the regular singular point is at}\hspace {3pt} u =0 \\ {} & {} & a_{1} u \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )+\left (b_{1} u -\frac {b_{1} a_{0}}{a_{1}}+b_{0} \right ) \left (\frac {d}{d u}y \left (u \right )\right )-b_{1} m y \left (u \right )=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \left (u \right ) \\ {} & {} & y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +r} \\ \square & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) u^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & u^{m}\cdot \left (\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +r \right ) u^{k +r} \\ {} & \circ & \textrm {Convert}\hspace {3pt} u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) \left (k +r -1\right ) u^{k +r -1} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & u \cdot \left (\frac {d}{d u}\frac {d}{d u}y \left (u \right )\right )=\moverset {\infty }{\munderset {k =-1}{\sum }}a_{k +1} \left (k +1+r \right ) \left (k +r \right ) u^{k +r} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & -\frac {a_{0} r \left (-a_{1}^{2} r +a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0} \right ) u^{-1+r}}{a_{1}}+\left (\moverset {\infty }{\munderset {k =0}{\sum }}\left (-\frac {a_{k +1} \left (k +1+r \right ) \left (-a_{1}^{2} \left (k +1\right )-a_{1}^{2} r +a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0} \right )}{a_{1}}+a_{k} b_{1} \left (k +r -m \right )\right ) u^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & -\frac {r \left (-a_{1}^{2} r +a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0} \right )}{a_{1}}=0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{0, \frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}\right \} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \frac {a_{k +1} \left (k +1+r \right ) \left (k +r \right ) a_{1}^{2}+\left (b_{0} \left (k +1+r \right ) a_{k +1}+a_{k} b_{1} \left (k +r -m \right )\right ) a_{1} -a_{0} b_{1} a_{k +1} \left (k +1+r \right )}{a_{1}}=0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +1}=\frac {a_{k} b_{1} \left (k +r -m \right ) a_{1}}{\left (k +1+r \right ) \left (-a_{1}^{2} k -a_{1}^{2} r +a_{0} b_{1} -a_{1} b_{0} \right )} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =0 \\ {} & {} & a_{k +1}=\frac {a_{k} b_{1} \left (k -m \right ) a_{1}}{\left (k +1\right ) \left (-a_{1}^{2} k +a_{0} b_{1} -a_{1} b_{0} \right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =0 \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k}, a_{k +1}=\frac {a_{k} b_{1} \left (k -m \right ) a_{1}}{\left (k +1\right ) \left (-a_{1}^{2} k +a_{0} b_{1} -a_{1} b_{0} \right )}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +\frac {a_{0}}{a_{1}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +\frac {a_{0}}{a_{1}}\right )^{k}, a_{k +1}=\frac {a_{k} b_{1} \left (k -m \right ) a_{1}}{\left (k +1\right ) \left (-a_{1}^{2} k +a_{0} b_{1} -a_{1} b_{0} \right )}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}} \\ {} & {} & a_{k +1}=\frac {a_{k} b_{1} \left (k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}-m \right ) a_{1}}{\left (k +1+\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}\right ) \left (-a_{1}^{2} k -a_{1}^{2}\right )} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}} \\ {} & {} & \left [y \left (u \right )=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} u^{k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}}, a_{k +1}=\frac {a_{k} b_{1} \left (k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}-m \right ) a_{1}}{\left (k +1+\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}\right ) \left (-a_{1}^{2} k -a_{1}^{2}\right )}\right ] \\ \bullet & {} & \textrm {Revert the change of variables}\hspace {3pt} u =x +\frac {a_{0}}{a_{1}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +\frac {a_{0}}{a_{1}}\right )^{k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}}, a_{k +1}=\frac {a_{k} b_{1} \left (k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}-m \right ) a_{1}}{\left (k +1+\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}\right ) \left (-a_{1}^{2} k -a_{1}^{2}\right )}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (x +\frac {a_{0}}{a_{1}}\right )^{k}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} \left (x +\frac {a_{0}}{a_{1}}\right )^{k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}}\right ), a_{k +1}=\frac {a_{k} b_{1} \left (k -m \right ) a_{1}}{\left (k +1\right ) \left (-a_{1}^{2} k +a_{0} b_{1} -a_{1} b_{0} \right )}, b_{k +1}=\frac {b_{k} b_{1} \left (k +\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}-m \right ) a_{1}}{\left (k +1+\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}\right ) \left (-a_{1}^{2} k -a_{1}^{2}\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 -> elliptic -> Legendre -> Kummer -> hyper3: Equivalence to 1F1 under a power @ Moebius <- hyper3 successful: received ODE is equivalent to the 1F1 ODE <- Kummer successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 101
dsolve((a__1*x+a__0)*diff(y(x),x$2)+(b__1*x+b__0)*diff(y(x),x)-m*b__1*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \left (a_{1} x +a_{0} \right )^{\frac {a_{0} b_{1} +a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}} {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \left (\operatorname {KummerM}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{1} +\operatorname {KummerU}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.278 (sec). Leaf size: 102
DSolve[(a1*x+a0)*y''[x]+(b1*x+b0)*y'[x]-m*b1*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {\text {b1} x}{\text {a1}}} (\text {a0}+\text {a1} x)^{\frac {\text {a0} \text {b1}+\text {a1}^2-\text {a1} \text {b0}}{\text {a1}^2}} \left (c_1 \operatorname {HypergeometricU}\left (m+1,-\frac {\text {b0}}{\text {a1}}+\frac {\text {a0} \text {b1}}{\text {a1}^2}+2,\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )+c_2 L_{-m-1}^{\frac {\text {a1}^2-\text {b0} \text {a1}+\text {a0} \text {b1}}{\text {a1}^2}}\left (\frac {\text {b1} (\text {a0}+\text {a1} x)}{\text {a1}^2}\right )\right ) \]