Internal problem ID [9548]
Internal file name [OUTPUT/8488_Monday_June_06_2022_03_12_48_AM_76375593/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1219.
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.
\[ \boxed {x^{2} y^{\prime \prime }+2 x f \left (x \right ) y^{\prime }+\left (f^{\prime }\left (x \right ) x +f \left (x \right )^{2}-f \left (x \right )+a \,x^{2}+b x +c \right ) y=0} \]
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 -> 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: 0.094 (sec). Leaf size: 69
dsolve(x^2*diff(diff(y(x),x),x)+2*x*f(x)*diff(y(x),x)+(x*diff(f(x),x)+f(x)^2-f(x)+a*x^2+b*x+c)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-\left (\int \frac {f \left (x \right )}{x}d x \right )} \left (\operatorname {WhittakerM}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right ) c_{1} +\operatorname {WhittakerW}\left (-\frac {i b}{2 \sqrt {a}}, \frac {\sqrt {1-4 c}}{2}, 2 i x \sqrt {a}\right ) c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.319 (sec). Leaf size: 151
DSolve[y[x]*(c + b*x + a*x^2 - f[x] + f[x]^2 + x*Derivative[1][f][x]) + 2*x*f[x]*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {i b}{\sqrt {a}}+\sqrt {1-4 c}+1\right ),\sqrt {1-4 c}+1,2 i \sqrt {a} x\right )+c_2 L_{\frac {1}{2} \left (-\frac {i b}{\sqrt {a}}-\sqrt {1-4 c}-1\right )}^{\sqrt {1-4 c}}\left (2 i \sqrt {a} x\right )\right ) \exp \left (\int _1^x\frac {-2 f(K[1])-2 i \sqrt {a} K[1]+\sqrt {1-4 c}+1}{2 K[1]}dK[1]\right ) \]