Internal problem ID [9353]
Internal file name [OUTPUT/8289_Monday_June_06_2022_02_38_42_AM_76317891/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1020.
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.
\[ \boxed {y^{\prime \prime }+\left (a \,{\mathrm e}^{2 x}+b \,{\mathrm e}^{x}+c \right ) y=0} \]
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 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 Change of variables used: [x = ln(t)] Linear ODE actually solved: (a*t^2+b*t+c)*u(t)+t*diff(u(t),t)+t^2*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.297 (sec). Leaf size: 58
dsolve(diff(diff(y(x),x),x)+(a*exp(2*x)+b*exp(x)+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\frac {x}{2}} \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, i \sqrt {c}, 2 i \sqrt {a}\, {\mathrm e}^{x}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.789 (sec). Leaf size: 136
DSolve[(c + b*E^x + a*E^(2*x))*y[x] + y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-i \sqrt {a} e^x} \left (e^x\right )^{i \sqrt {c}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {i b}{2 \sqrt {a}}+i \sqrt {c}+\frac {1}{2},2 i \sqrt {c}+1,2 i \sqrt {a} e^x\right )+c_2 L_{-\frac {i b}{2 \sqrt {a}}-i \sqrt {c}-\frac {1}{2}}^{2 i \sqrt {c}}\left (2 i \sqrt {a} e^x\right )\right ) \]