60.4.50 problem 1498
Internal
problem
ID
[11501]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
3,
linear
third
order
Problem
number
:
1498
Date
solved
:
Tuesday, January 28, 2025 at 06:06:39 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{2} y^{\prime \prime \prime }-2 \left (n +1\right ) x y^{\prime \prime }+\left (a \,x^{2}+6 n \right ) y^{\prime }-2 a x y&=0 \end{align*}
✓ Solution by Maple
Time used: 0.035 (sec). Leaf size: 53
dsolve(x^2*diff(diff(diff(y(x),x),x),x)-2*(n+1)*x*diff(diff(y(x),x),x)+(a*x^2+6*n)*diff(y(x),x)-2*y(x)*a*x=0,y(x), singsol=all)
\[
y = c_{1} x^{n +\frac {1}{2}} \operatorname {BesselJ}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_{2} x^{n +\frac {1}{2}} \operatorname {BesselY}\left (-n -\frac {1}{2}, \sqrt {a}\, x \right )+c_3 \left (a \,x^{2}+4 n -2\right )
\]
✓ Solution by Mathematica
Time used: 5.575 (sec). Leaf size: 353
DSolve[-2*a*x*y[x] + (6*n + a*x^2)*D[y[x],x] - 2*(1 + n)*x*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[
y(x)\to 2^{-n-\frac {3}{2}} \left (\pi c_3 4^n x^4 \sec (\pi n) \operatorname {Gamma}\left (\frac {3}{2}-n\right ) \left (\sqrt {a} x\right )^{-n-\frac {1}{2}} \operatorname {BesselJ}\left (n+\frac {1}{2},\sqrt {a} x\right ) \, _1\tilde {F}_2\left (\frac {3}{2}-n;\frac {1}{2}-n,\frac {5}{2}-n;-\frac {a x^2}{4}\right )+\frac {\operatorname {BesselY}\left (n+\frac {1}{2},\sqrt {a} x\right ) \left (2 \pi c_3 \left (4 n^2-1\right ) \left (\sqrt {a} x\right )^{n+\frac {1}{2}}+a 2^{n+\frac {1}{2}} \operatorname {Gamma}\left (n+\frac {3}{2}\right ) \left (2 a c_2 x^{n+\frac {1}{2}}-\pi \sqrt {a} c_3 x^3 \operatorname {BesselJ}\left (n-\frac {1}{2},\sqrt {a} x\right )-2 \pi c_3 x^2 \operatorname {BesselJ}\left (n-\frac {3}{2},\sqrt {a} x\right )\right )\right )+\operatorname {BesselJ}\left (n+\frac {1}{2},\sqrt {a} x\right ) \left (2 \pi c_3 \left (4 n^2-1\right ) \tan (\pi n) \left (\sqrt {a} x\right )^{n+\frac {1}{2}}+a 2^{n+\frac {1}{2}} \operatorname {Gamma}\left (n+\frac {3}{2}\right ) \left (2 a c_1 x^{n+\frac {1}{2}}-\pi \sqrt {a} c_3 x^3 \tan (\pi n) \operatorname {BesselJ}\left (n-\frac {1}{2},\sqrt {a} x\right )-2 \pi c_3 x^2 \tan (\pi n) \operatorname {BesselJ}\left (n-\frac {3}{2},\sqrt {a} x\right )\right )\right )}{a^2 \operatorname {Gamma}\left (n+\frac {3}{2}\right )}\right )
\]