Internal problem ID [9671]
Internal file name [OUTPUT/8613_Monday_June_06_2022_04_28_07_AM_51958275/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1344.
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.
\[ \boxed {y^{\prime \prime }+\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}=0} \]
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{4}+\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\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 ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}} \\ \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 ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y}{x^{4}}=0 \\ \bullet & {} & \textrm {Multiply by denominators of the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime }\right ) x^{4}+\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y=0 \\ \bullet & {} & \textrm {Make a change of variables}\hspace {3pt} \\ {} & {} & t =\ln \left (x \right ) \\ \square & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {1st}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & y^{\prime }=\left (\frac {d}{d t}y \left (t \right )\right ) t^{\prime }\left (x \right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\frac {d}{d t}y \left (t \right )}{x} \\ {} & \circ & \textrm {Calculate the}\hspace {3pt} \hspace {3pt}\textrm {2nd}\hspace {3pt} \hspace {3pt}\textrm {derivative of}\hspace {3pt} \hspace {3pt}\textrm {y}\hspace {3pt} \hspace {3pt}\textrm {with respect to}\hspace {3pt} \hspace {3pt}\textrm {x}\hspace {3pt} \hspace {3pt}\textrm {, using the chain rule}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right ) {t^{\prime }\left (x \right )}^{2}+\left (\frac {d}{d x}t^{\prime }\left (x \right )\right ) \left (\frac {d}{d t}y \left (t \right )\right ) \\ {} & \circ & \textrm {Compute derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y^{\prime }=\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}} \\ & {} & \textrm {Substitute the change of variables back into the ODE}\hspace {3pt} \\ {} & {} & \left (\frac {\frac {d}{d t}\frac {d}{d t}y \left (t \right )}{x^{2}}-\frac {\frac {d}{d t}y \left (t \right )}{x^{2}}\right ) x^{4}+\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y \left (t \right )=0 \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & x^{2} \left (\frac {d}{d t}\frac {d}{d t}y \left (t \right )\right )-x^{2} \left (\frac {d}{d t}y \left (t \right )\right )-y \left (t \right ) v^{2}+{\mathrm e}^{\frac {2}{x}} y \left (t \right )=0 \\ \bullet & {} & \textrm {Isolate 2nd derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d t}\frac {d}{d t}y \left (t \right )=-\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y \left (t \right )}{x^{2}}+\frac {d}{d t}y \left (t \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y \left (t \right )\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 t}\frac {d}{d t}y \left (t \right )-\frac {d}{d t}y \left (t \right )+\frac {\left ({\mathrm e}^{\frac {2}{x}}-v^{2}\right ) y \left (t \right )}{x^{2}}=0 \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}-r +\frac {{\mathrm e}^{\frac {2}{x}}-v^{2}}{x^{2}}=0 \\ \bullet & {} & \textrm {Factor the characteristic polynomial}\hspace {3pt} \\ {} & {} & \frac {r^{2} x^{2}-r \,x^{2}-v^{2}+{\mathrm e}^{\frac {2}{x}}}{x^{2}}=0 \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (\frac {\frac {x}{2}+\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}}{x}, \frac {\frac {x}{2}-\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}}{x}\right ) \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )={\mathrm e}^{\frac {\left (\frac {x}{2}+\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {2nd solution of the ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )={\mathrm e}^{\frac {\left (\frac {x}{2}-\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y \left (t \right )=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y \left (t \right )=c_{1} {\mathrm e}^{\frac {\left (\frac {x}{2}+\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) t}{x}}+c_{2} {\mathrm e}^{\frac {\left (\frac {x}{2}-\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) t}{x}} \\ \bullet & {} & \textrm {Change variables back using}\hspace {3pt} t =\ln \left (x \right ) \\ {} & {} & y=c_{1} {\mathrm e}^{\frac {\left (\frac {x}{2}+\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) \ln \left (x \right )}{x}}+c_{2} {\mathrm e}^{\frac {\left (\frac {x}{2}-\frac {\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2}\right ) \ln \left (x \right )}{x}} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=c_{1} x^{\frac {x +\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2 x}}+c_{2} x^{-\frac {-x +\sqrt {x^{2}-4 \,{\mathrm e}^{\frac {2}{x}}+4 v^{2}}}{2 x}} \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler 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 <- Bessel successful <- special function solution successful Change of variables used: [x = 1/ln(t)] Linear ODE actually solved: (-ln(t)*v^2+ln(t)*t^2)*u(t)+(t*ln(t)+2*t)*diff(u(t),t)+ln(t)*t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.156 (sec). Leaf size: 23
dsolve(diff(diff(y(x),x),x) = -(exp(2/x)-v^2)/x^4*y(x),y(x), singsol=all)
\[ y \left (x \right ) = x \left (c_{1} \operatorname {BesselJ}\left (v , {\mathrm e}^{\frac {1}{x}}\right )+c_{2} \operatorname {BesselY}\left (v , {\mathrm e}^{\frac {1}{x}}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.724 (sec). Leaf size: 100
DSolve[y''[x] == -(((E^(2/x) - v^2)*y[x])/x^4),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {(-1)^{-v} 2^{\frac {3 v}{2}+\frac {1}{2}} \left (-e^{2/x}\right )^{-v/2} \left (e^{2/x}\right )^{v/2} \left (c_1 (-1)^v \operatorname {BesselI}\left (v,\sqrt {-e^{2/x}}\right )+c_2 K_v\left (\sqrt {-e^{2/x}}\right )\right )}{\log \left (e^{2/x}\right )} \]