Internal problem ID [9624]
Internal file name [OUTPUT/8565_Monday_June_06_2022_03_52_51_AM_76047978/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1296.
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.
\[ \boxed {\operatorname {a2} \,x^{2} y^{\prime \prime }+\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x \right ) y^{\prime }+\left (\operatorname {a0} \,x^{2}+\operatorname {b0} x +\operatorname {c0} \right ) y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \mathit {a2} \,x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (\mathit {a1} x +\mathit {b1} \right ) y^{\prime }+\left (\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0} \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 (\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0} \right ) y}{\mathit {a2} \,x^{2}}-\frac {\left (\mathit {a1} x +\mathit {b1} \right ) y^{\prime }}{x \mathit {a2}} \\ \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 (\mathit {a1} x +\mathit {b1} \right ) y^{\prime }}{x \mathit {a2}}+\frac {\left (\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0} \right ) y}{\mathit {a2} \,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 {\mathit {a1} x +\mathit {b1}}{\mathit {a2} x}, P_{3}\left (x \right )=\frac {\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0}}{\mathit {a2} \,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 {\mathit {b1}}{\mathit {a2}} \\ {} & \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 {\mathit {c0}}{\mathit {a2}} \\ {} & \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} \\ {} & {} & \mathit {a2} \,x^{2} \left (\frac {d}{d x}y^{\prime }\right )+x \left (\mathit {a1} x +\mathit {b1} \right ) y^{\prime }+\left (\mathit {a0} \,x^{2}+\mathit {b0} x +\mathit {c0} \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^{m}\cdot y^{\prime }\hspace {3pt}\textrm {to series expansion for}\hspace {3pt} m =1..2 \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} \left (k +r \right ) x^{k +r -1+m} \\ {} & \circ & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +1-m \\ {} & {} & x^{m}\cdot y^{\prime }=\moverset {\infty }{\munderset {k =-1+m}{\sum }}a_{k +1-m} \left (k +1-m +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 (\mathit {a2} \,r^{2}-\mathit {a2} r +\mathit {b1} r +\mathit {c0} \right ) x^{r}+\left (\left (\mathit {a2} \,r^{2}+\mathit {a2} r +\mathit {b1} r +\mathit {b1} +\mathit {c0} \right ) a_{1}+a_{0} \left (\mathit {a1} r +\mathit {b0} \right )\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (\mathit {a2} \,k^{2}+2 \mathit {a2} k r +\mathit {a2} \,r^{2}-\mathit {a2} k -\mathit {a2} r +\mathit {b1} k +\mathit {b1} r +\mathit {c0} \right )+a_{k -1} \left (\mathit {a1} \left (k -1\right )+\mathit {a1} r +\mathit {b0} \right )+a_{k -2} \mathit {a0} \right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & \mathit {a2} \,r^{2}-\mathit {a2} r +\mathit {b1} r +\mathit {c0} =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}, -\frac {-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (\mathit {a2} \,r^{2}+\mathit {a2} r +\mathit {b1} r +\mathit {b1} +\mathit {c0} \right ) a_{1}+a_{0} \left (\mathit {a1} r +\mathit {b0} \right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {a_{0} \left (\mathit {a1} r +\mathit {b0} \right )}{\mathit {a2} \,r^{2}+\mathit {a2} r +\mathit {b1} r +\mathit {b1} +\mathit {c0}} \\ \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 ) \mathit {a2} +\mathit {b1} k +\mathit {b1} r +\mathit {c0} \right ) a_{k}+\mathit {a1} k a_{k -1}+\mathit {a1} r a_{k -1}+\left (-\mathit {a1} +\mathit {b0} \right ) a_{k -1}+a_{k -2} \mathit {a0} =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2+r \right ) \left (k +1+r \right ) \mathit {a2} +\mathit {b1} \left (k +2\right )+\mathit {b1} r +\mathit {c0} \right ) a_{k +2}+\mathit {a1} \left (k +2\right ) a_{k +1}+\mathit {a1} r a_{k +1}+\left (-\mathit {a1} +\mathit {b0} \right ) a_{k +1}+a_{k} \mathit {a0} =0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {\mathit {a1} k a_{k +1}+\mathit {a1} r a_{k +1}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}+2 \mathit {a2} k r +\mathit {a2} \,r^{2}+3 \mathit {a2} k +3 \mathit {a2} r +\mathit {b1} k +\mathit {b1} r +2 \mathit {a2} +2 \mathit {b1} +\mathit {c0}} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}} \\ {} & {} & a_{k +2}=-\frac {\mathit {a1} k a_{k +1}+\frac {\mathit {a1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) a_{k +1}}{2 \mathit {a2}}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}+k \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}+\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k +\frac {\mathit {b1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}}, a_{k +2}=-\frac {\mathit {a1} k a_{k +1}+\frac {\mathit {a1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) a_{k +1}}{2 \mathit {a2}}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}+k \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}+\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k +\frac {\mathit {b1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}, a_{1}=-\frac {a_{0} \left (\frac {\mathit {a1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {b0} \right )}{\frac {\left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+\frac {\mathit {a2}}{2}+\frac {\mathit {b1}}{2}+\frac {\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\frac {\mathit {b1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =-\frac {-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}} \\ {} & {} & a_{k +2}=-\frac {\mathit {a1} k a_{k +1}-\frac {\mathit {a1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) a_{k +1}}{2 \mathit {a2}}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}-k \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}-\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k -\frac {\mathit {b1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =-\frac {-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k -\frac {-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}}, a_{k +2}=-\frac {\mathit {a1} k a_{k +1}-\frac {\mathit {a1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) a_{k +1}}{2 \mathit {a2}}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}-k \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}-\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k -\frac {\mathit {b1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}, a_{1}=-\frac {a_{0} \left (-\frac {\mathit {a1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {b0} \right )}{\frac {\left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+\frac {\mathit {a2}}{2}+\frac {\mathit {b1}}{2}-\frac {\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}-\frac {\mathit {b1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}b_{k} x^{k -\frac {-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2 \mathit {a2}}}\right ), a_{k +2}=-\frac {\mathit {a1} k a_{k +1}+\frac {\mathit {a1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) a_{k +1}}{2 \mathit {a2}}+a_{k} \mathit {a0} +\mathit {a1} a_{k +1}+\mathit {b0} a_{k +1}}{\mathit {a2} \,k^{2}+k \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}+\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k +\frac {\mathit {b1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}, a_{1}=-\frac {a_{0} \left (\frac {\mathit {a1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {b0} \right )}{\frac {\left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+\frac {\mathit {a2}}{2}+\frac {\mathit {b1}}{2}+\frac {\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\frac {\mathit {b1} \left (\mathit {a2} -\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}, b_{k +2}=-\frac {\mathit {a1} k b_{k +1}-\frac {\mathit {a1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right ) b_{k +1}}{2 \mathit {a2}}+b_{k} \mathit {a0} +\mathit {a1} b_{k +1}+\mathit {b0} b_{k +1}}{\mathit {a2} \,k^{2}-k \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )+\frac {\left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+3 \mathit {a2} k +\frac {7 \mathit {a2}}{2}+\frac {\mathit {b1}}{2}-\frac {3 \sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}+\mathit {b1} k -\frac {\mathit {b1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}, b_{1}=-\frac {b_{0} \left (-\frac {\mathit {a1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {b0} \right )}{\frac {\left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )^{2}}{4 \mathit {a2}}+\frac {\mathit {a2}}{2}+\frac {\mathit {b1}}{2}-\frac {\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}}{2}-\frac {\mathit {b1} \left (-\mathit {a2} +\mathit {b1} +\sqrt {\mathit {a2}^{2}-2 \mathit {a2} \mathit {b1} -4 \mathit {c0} \mathit {a2} +\mathit {b1}^{2}}\right )}{2 \mathit {a2}}+\mathit {c0}}\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: 0.141 (sec). Leaf size: 150
dsolve(a2*x^2*diff(diff(y(x),x),x)+(a1*x^2+b1*x)*diff(y(x),x)+(a0*x^2+b0*x+c0)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {\operatorname {a1} x}{2 \operatorname {a2}}} x^{-\frac {\operatorname {b1}}{2 \operatorname {a2}}} \left (\operatorname {WhittakerM}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right ) c_{1} +c_{2} \operatorname {WhittakerW}\left (-\frac {\operatorname {a1} \operatorname {b1} -2 \operatorname {a2} \operatorname {b0}}{2 \operatorname {a2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, \frac {\sqrt {\operatorname {a2}^{2}+\left (-2 \operatorname {b1} -4 \operatorname {c0} \right ) \operatorname {a2} +\operatorname {b1}^{2}}}{2 \operatorname {a2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, x}{\operatorname {a2}}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.331 (sec). Leaf size: 272
DSolve[(c0 + b0*x + a0*x^2)*y[x] + (b1*x + a1*x^2)*y'[x] + a2*x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {x \left (\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}+\text {a1}\right )}{2 \text {a2}}} x^{\frac {\sqrt {\text {a2}^2-2 \text {a2} (\text {b1}+2 \text {c0})+\text {b1}^2}+\text {a2}-\text {b1}}{2 \text {a2}}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}},\frac {\text {a2}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}},\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )+c_2 L_{-\frac {-\frac {2 \text {b0} \text {a2}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\text {a2}+\frac {\text {a1} \text {b1}}{\sqrt {\text {a1}^2-4 \text {a0} \text {a2}}}+\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{2 \text {a2}}}^{\frac {\sqrt {\text {a2}^2-2 (\text {b1}+2 \text {c0}) \text {a2}+\text {b1}^2}}{\text {a2}}}\left (\frac {\sqrt {\text {a1}^2-4 \text {a0} \text {a2}} x}{\text {a2}}\right )\right ) \]