Internal problem ID [9544]
Internal file name [OUTPUT/8484_Monday_June_06_2022_03_11_59_AM_92131841/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1215.
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.
\[ \boxed {x^{2} y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\operatorname {a1} \,x^{2 n}+\operatorname {b1} \,x^{n}+\operatorname {c1} \right ) y=0} \]
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 an equivalence, under non-integer power transformations, to LODEs admitting Liouvillian solutions. -> 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: 0.25 (sec). Leaf size: 148
dsolve(x^2*diff(diff(y(x),x),x)+(a*x^n+b)*diff(y(x),x)*x+(a1*x^(2*n)+b1*x^n+c1)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \left (c_{1} \operatorname {WhittakerM}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )+c_{2} \operatorname {WhittakerW}\left (-\frac {\left (b +n -1\right ) a -2 \operatorname {b1}}{2 \sqrt {a^{2}-4 \operatorname {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \operatorname {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \operatorname {a1}}\, x^{n}}{n}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.342 (sec). Leaf size: 412
DSolve[(c1 + b1*x^n + a1*x^(2*n))*y[x] + x*(b + a*x^n)*y'[x] + x^2*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to x^{\frac {1}{2}-\frac {n}{2}} 2^{\frac {1}{2} \left (\frac {\sqrt {n^2 \left (b^2-2 b-4 \text {c1}+1\right )}}{n^2}+1\right )} e^{-\frac {\left (\sqrt {a^2-4 \text {a1}}+a\right ) x^n}{2 n}} \left (x^n\right )^{\frac {\sqrt {n^2 \left (b^2-2 b-4 \text {c1}+1\right )}-b n+n^2}{2 n^2}} \left (c_1 \operatorname {HypergeometricU}\left (\frac {\left (n^2+\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}\right ) a^2+\sqrt {a^2-4 \text {a1}} n (b+n-1) a-2 \sqrt {a^2-4 \text {a1}} \text {b1} n-4 \text {a1} \left (n^2+\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}\right )}{2 \left (a^2-4 \text {a1}\right ) n^2},\frac {n^2+\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}}{n^2},\frac {\sqrt {a^2-4 \text {a1}} x^n}{n}\right )+c_2 L_{-\frac {\left (n^2+\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}\right ) a^2+\sqrt {a^2-4 \text {a1}} n (b+n-1) a-2 \sqrt {a^2-4 \text {a1}} \text {b1} n-4 \text {a1} \left (n^2+\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}\right )}{2 \left (a^2-4 \text {a1}\right ) n^2}}^{\frac {\sqrt {\left (b^2-2 b-4 \text {c1}+1\right ) n^2}}{n^2}}\left (\frac {\sqrt {a^2-4 \text {a1}} x^n}{n}\right )\right ) \]