Internal problem ID [9536]
Internal file name [OUTPUT/8476_Monday_June_06_2022_03_10_33_AM_35120029/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1207.
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.
\[ \boxed {x^{2} y^{\prime \prime }+\left (a x +b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2}+\operatorname {b1} x +\operatorname {c1} \right ) y=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (a x +b \right ) y^{\prime } x +\left (\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \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 {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \right ) y}{x^{2}}-\frac {\left (a x +b \right ) y^{\prime }}{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 {\left (a x +b \right ) y^{\prime }}{x}+\frac {\left (\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \right ) y}{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 {a x +b}{x}, P_{3}\left (x \right )=\frac {\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1}}{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}}}=b \\ {} & \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}}}=\mathit {c1} \\ {} & \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} \\ {} & {} & x^{2} \left (\frac {d}{d x}y^{\prime }\right )+\left (a x +b \right ) y^{\prime } x +\left (\mathit {a1} \,x^{2}+\mathit {b1} x +\mathit {c1} \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 (b r +r^{2}+\mathit {c1} -r \right ) x^{r}+\left (\left (b r +r^{2}+b +\mathit {c1} +r \right ) a_{1}+a_{0} \left (a r +\mathit {b1} \right )\right ) x^{1+r}+\left (\moverset {\infty }{\munderset {k =2}{\sum }}\left (a_{k} \left (b k +b r +k^{2}+2 k r +r^{2}+\mathit {c1} -k -r \right )+a_{k -1} \left (a \left (k -1\right )+a r +\mathit {b1} \right )+a_{k -2} \mathit {a1} \right ) x^{k +r}\right )=0 \\ \bullet & {} & a_{0}\textrm {cannot be 0 by assumption, giving the indicial equation}\hspace {3pt} \\ {} & {} & b r +r^{2}+\mathit {c1} -r =0 \\ \bullet & {} & \textrm {Values of r that satisfy the indicial equation}\hspace {3pt} \\ {} & {} & r \in \left \{\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}, \frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right \} \\ \bullet & {} & \textrm {Each term must be 0}\hspace {3pt} \\ {} & {} & \left (b r +r^{2}+b +\mathit {c1} +r \right ) a_{1}+a_{0} \left (a r +\mathit {b1} \right )=0 \\ \bullet & {} & \textrm {Solve for the dependent coefficient(s)}\hspace {3pt} \\ {} & {} & a_{1}=-\frac {a_{0} \left (a r +\mathit {b1} \right )}{b r +r^{2}+b +\mathit {c1} +r} \\ \bullet & {} & \textrm {Each term in the series must be 0, giving the recursion relation}\hspace {3pt} \\ {} & {} & \left (k^{2}+\left (b +2 r -1\right ) k +r^{2}+\left (-1+b \right ) r +\mathit {c1} \right ) a_{k}+a k a_{k -1}+a r a_{k -1}+\left (-a +\mathit {b1} \right ) a_{k -1}+a_{k -2} \mathit {a1} =0 \\ \bullet & {} & \textrm {Shift index using}\hspace {3pt} k \mathrm {->}k +2 \\ {} & {} & \left (\left (k +2\right )^{2}+\left (b +2 r -1\right ) \left (k +2\right )+r^{2}+\left (-1+b \right ) r +\mathit {c1} \right ) a_{k +2}+a \left (k +2\right ) a_{k +1}+a r a_{k +1}+\left (-a +\mathit {b1} \right ) a_{k +1}+a_{k} \mathit {a1} =0 \\ \bullet & {} & \textrm {Recursion relation that defines series solution to ODE}\hspace {3pt} \\ {} & {} & a_{k +2}=-\frac {a k a_{k +1}+a r a_{k +1}+a a_{k +1}+a_{k} \mathit {a1} +\mathit {b1} a_{k +1}}{b k +b r +k^{2}+2 k r +r^{2}+2 b +\mathit {c1} +3 k +3 r +2} \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2} \\ {} & {} & a_{k +2}=-\frac {a k a_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) a_{k +1}+a a_{k +1}+a_{k} \mathit {a1} +\mathit {b1} a_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}-\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, a_{k +2}=-\frac {a k a_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) a_{k +1}+a a_{k +1}+a_{k} \mathit {a1} +\mathit {b1} a_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}-\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, a_{1}=-\frac {a_{0} \left (a \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+\mathit {b1} \right )}{b \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +\frac {1}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}\right ] \\ \bullet & {} & \textrm {Recursion relation for}\hspace {3pt} r =\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2} \\ {} & {} & a_{k +2}=-\frac {a k a_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) a_{k +1}+a a_{k +1}+a_{k} \mathit {a1} +\mathit {b1} a_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}+\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}} \\ \bullet & {} & \textrm {Solution for}\hspace {3pt} r =\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2} \\ {} & {} & \left [y=\moverset {\infty }{\munderset {k =0}{\sum }}a_{k} x^{k +\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, a_{k +2}=-\frac {a k a_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) a_{k +1}+a a_{k +1}+a_{k} \mathit {a1} +\mathit {b1} a_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}+\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, a_{1}=-\frac {a_{0} \left (a \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+\mathit {b1} \right )}{b \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +\frac {1}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}\right ] \\ \bullet & {} & \textrm {Combine solutions and rename parameters}\hspace {3pt} \\ {} & {} & \left [y=\left (\moverset {\infty }{\munderset {k =0}{\sum }}c_{k} x^{k +\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}\right )+\left (\moverset {\infty }{\munderset {k =0}{\sum }}d_{k} x^{k +\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}\right ), c_{k +2}=-\frac {a k c_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) c_{k +1}+a c_{k +1}+c_{k} \mathit {a1} +\mathit {b1} c_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}-\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, c_{1}=-\frac {c_{0} \left (a \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+\mathit {b1} \right )}{b \left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +\frac {1}{2}-\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, d_{k +2}=-\frac {a k d_{k +1}+a \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right ) d_{k +1}+a d_{k +1}+d_{k} \mathit {a1} +\mathit {b1} d_{k +1}}{b k +b \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+k^{2}+2 k \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +3 k +\frac {7}{2}+\frac {3 \sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}, d_{1}=-\frac {d_{0} \left (a \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+\mathit {b1} \right )}{b \left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )+{\left (\frac {1}{2}-\frac {b}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}\right )}^{2}+\frac {b}{2}+\mathit {c1} +\frac {1}{2}+\frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2}}\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: 0.125 (sec). Leaf size: 110
dsolve(x^2*diff(diff(y(x),x),x)+(a*x+b)*diff(y(x),x)*x+(a1*x^2+b1*x+c1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {a x}{2}} x^{-\frac {b}{2}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )+c_{2} \operatorname {WhittakerW}\left (-\frac {a b -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2}, \sqrt {a^{2}-4 \operatorname {a1}}\, x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.177 (sec). Leaf size: 223
DSolve[(c1 + b1*x + a1*x^2)*y[x] + x*(b + a*x)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-\frac {1}{2} x \left (\sqrt {a^2-4 \text {a1}}+a\right )} x^{\frac {1}{2} \left (\sqrt {b^2-2 b-4 \text {c1}+1}-b+1\right )} \left (c_1 \operatorname {HypergeometricU}\left (\frac {a b-2 \text {b1}+\sqrt {a^2-4 \text {a1}} \left (\sqrt {b^2-2 b-4 \text {c1}+1}+1\right )}{2 \sqrt {a^2-4 \text {a1}}},\sqrt {b^2-2 b-4 \text {c1}+1}+1,\sqrt {a^2-4 \text {a1}} x\right )+c_2 L_{\frac {-a b+2 \text {b1}-\sqrt {a^2-4 \text {a1}} \left (\sqrt {b^2-2 b-4 \text {c1}+1}+1\right )}{2 \sqrt {a^2-4 \text {a1}}}}^{\sqrt {b^2-2 b-4 \text {c1}+1}}\left (\sqrt {a^2-4 \text {a1}} x\right )\right ) \]