60.4.54 problem 1502

Internal problem ID [11505]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1502
Date solved : Tuesday, January 28, 2025 at 06:06:40 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-\left (x^{4}-6 x \right ) y^{\prime \prime }-\left (2 x^{3}-6\right ) y^{\prime }+2 x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.270 (sec). Leaf size: 103

dsolve(x^2*diff(diff(diff(y(x),x),x),x)-(x^4-6*x)*diff(diff(y(x),x),x)-(2*x^3-6)*diff(y(x),x)+2*x^2*y(x)=0,y(x), singsol=all)
 

\[ y = \frac {c_3 \left (\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}-\operatorname {BesselK}\left (\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselK}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )+c_{2} \left (\int {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \left (\operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right ) x^{3}+\operatorname {BesselI}\left (-\frac {5}{6}, -\frac {x^{3}}{6}\right ) x^{3}-2 \operatorname {BesselI}\left (\frac {1}{6}, -\frac {x^{3}}{6}\right )\right )d x \right )+c_{1}}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.062 (sec). Leaf size: 72

DSolve[2*x^2*y[x] - (-6 + 2*x^3)*D[y[x],x] - (-6*x + x^4)*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {c_2 \int _1^x\operatorname {Hypergeometric1F1}\left (-\frac {2}{3},\frac {2}{3},\frac {K[1]^3}{3}\right )dK[1]+c_3 \int _1^x\sqrt [3]{-\frac {1}{3}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{3},\frac {4}{3},\frac {K[2]^3}{3}\right ) K[2]dK[2]+c_1}{x^2} \]