Internal problem ID [9418]
Internal file name [OUTPUT/8358_Monday_June_06_2022_02_50_01_AM_8077675/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1089.
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.
\[ \boxed {a y^{\prime \prime }-\left (a b +c +x \right ) y^{\prime }+\left (b \left (x +c \right )+d \right ) y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & a \left (\frac {d}{d x}y^{\prime }\right )+\left (-a b -c -x \right ) y^{\prime }+\left (b \left (x +c \right )+d \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 (b c +b x +d \right ) y}{a}+\frac {\left (a b +c +x \right ) y^{\prime }}{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 {\left (a b +c +x \right ) y^{\prime }}{a}+\frac {\left (b c +b x +d \right ) y}{a}=0 \\ \bullet & {} & \textrm {Multiply by denominators}\hspace {3pt} \\ {} & {} & a \left (\frac {d}{d x}y^{\prime }\right )+\left (-a b -c -x \right ) y^{\prime }+\left (b c +b x +d \right ) y=0 \\ \bullet & {} & \textrm {Assume series solution for}\hspace {3pt} y \\ {} & {} & y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k} \\ \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..1 \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )}{\sum }}a_{k} x^{k +m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k -m \\ {} & {} & x^{m}\cdot y=\moverset {\infty }{\munderset {k =\max \left (0, -m \right )+m}{\sum }}a_{k -m} x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} x^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =0..1 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =\max \left (0, 1-m \right )}{\sum }}a_{k} k \,x^{k -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =\max \left (0, 1-m \right )+m -1}{\sum }}a_{k +1-m} \left (k +1-m \right ) x^{k} \\ {} & \circ & \textrm {Convert}\hspace {3pt} \frac {d}{d x}y^{\prime }\hspace {3pt}\textrm {to series expansion}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\moverset {\infty }{\munderset {k =2}{\sum }}a_{k} k \left (k -1\right ) x^{k -2} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \frac {d}{d x}y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k +2} \left (k +2\right ) \left (k +1\right ) x^{k} \\ & {} & \textrm {Rewrite ODE with series expansions}\hspace {3pt} \\ {} & {} & 2 a a_{2}-a_{1} \left (a b +c \right )+a_{0} \left (b c +d \right )+\left (\moverset {\infty }{\munderset {k =1}{\sum }}\left (a a_{k +2} \left (k +2\right ) \left (k +1\right )-a_{k +1} \left (k +1\right ) \left (a b +c \right )+a_{k} \left (b c +d -k \right )+a_{k -1} b \right ) x^{k}\right )=0 \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & 2 a a_{2}-a_{1} \left (a b +c \right )+a_{0} \left (b c +d \right )=0 \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & -\left (k +1\right ) \left (b a_{k +1}-k a_{k +2}-2 a_{k +2}\right ) a +\left (-c a_{k +1}-a_{k}\right ) k -c a_{k +1}+\left (c a_{k}+a_{k -1}\right ) b +a_{k} d =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1 \\ {} & {} & -\left (k +2\right ) \left (b a_{k +2}-\left (k +1\right ) a_{k +3}-2 a_{k +3}\right ) a +\left (-c a_{k +2}-a_{k +1}\right ) \left (k +1\right )-c a_{k +2}+\left (c a_{k +1}+a_{k}\right ) b +a_{k +1} d =0 \\ \bullet & {} & \textrm {Recursion relation that defines the series solution to the ODE}\hspace {3pt} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k}, a_{k +3}=\frac {a b k a_{k +2}+2 a b a_{k +2}-b c a_{k +1}+c k a_{k +2}-b a_{k}+2 c a_{k +2}-a_{k +1} d +k a_{k +1}+a_{k +1}}{\left (k +2\right ) \left (k +3\right ) a}, 2 a a_{2}-a_{1} \left (a b +c \right )+a_{0} \left (b c +d \right )=0\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 -> hypergeometric -> heuristic approach <- heuristic approach successful <- hypergeometric successful <- special function solution successful`
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 58
dsolve(a*diff(diff(y(x),x),x)-(a*b+c+x)*diff(y(x),x)+(b*(x+c)+d)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{b x} \left (\operatorname {KummerM}\left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right ) c_{1} +\operatorname {KummerU}\left (-\frac {d}{2}, \frac {1}{2}, \frac {\left (a b -c -x \right )^{2}}{2 a}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.065 (sec). Leaf size: 63
DSolve[(d + b*(c + x))*y[x] - (a*b + c + x)*y'[x] + a*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{b x} \left (c_1 \operatorname {HermiteH}\left (d,\frac {-a b+c+x}{\sqrt {2} \sqrt {a}}\right )+c_2 \operatorname {Hypergeometric1F1}\left (-\frac {d}{2},\frac {1}{2},\frac {(-a b+c+x)^2}{2 a}\right )\right ) \]