Internal problem ID [8795]
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.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {y^{\prime \prime } x^{2}+\left (a \,x^{n}+b \right ) y^{\prime } x +\left (\mathit {a1} \,x^{2 n}+\mathit {b1} \,x^{n}+\mathit {c1} \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 167
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)
\[ y \relax (x ) = c_{1} x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \WhittakerM \left (-\frac {\left (b +n -1\right ) a -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \mathit {a1}}\, x^{n}}{n}\right )+c_{2} x^{-\frac {b}{2}-\frac {n}{2}+\frac {1}{2}} {\mathrm e}^{-\frac {a \,x^{n}}{2 n}} \WhittakerW \left (-\frac {\left (b +n -1\right ) a -2 \mathit {b1}}{2 \sqrt {a^{2}-4 \mathit {a1}}\, n}, \frac {\sqrt {b^{2}-2 b -4 \mathit {c1} +1}}{2 n}, \frac {\sqrt {a^{2}-4 \mathit {a1}}\, x^{n}}{n}\right ) \]
✓ Solution by Mathematica
Time used: 0.075 (sec). Leaf size: 325
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]
\begin{align*} y(x)\to x^{\frac {1}{2}-\frac {n}{2}} 2^{\frac {\sqrt {n^2 \left ((b-1)^2-4 \text {c1}\right )}}{2 n^2}+\frac {1}{2}} 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-1)^2-4 \text {c1}\right )}-b n+n^2}{2 n^2}} \left (c_1 \text {HypergeometricU}\left (\frac {\sqrt {a^2-4 \text {a1}} \left (\sqrt {n^2 \left ((b-1)^2-4 \text {c1}\right )}+n^2\right )+a n (b+n-1)-2 \text {b1} n}{2 n^2 \sqrt {a^2-4 \text {a1}}},\frac {\sqrt {n^2 \left ((b-1)^2-4 \text {c1}\right )}}{n^2}+1,\frac {\sqrt {a^2-4 \text {a1}} x^n}{n}\right )+c_2 L_{\frac {2 \text {b1} n-a (b+n-1) n-\sqrt {a^2-4 \text {a1}} \left (n^2+\sqrt {\left ((b-1)^2-4 \text {c1}\right ) n^2}\right )}{2 \sqrt {a^2-4 \text {a1}} n^2}}^{\frac {\sqrt {\left ((b-1)^2-4 \text {c1}\right ) n^2}}{n^2}}\left (\frac {\sqrt {a^2-4 \text {a1}} x^n}{n}\right )\right ) \\ \end{align*}