60.4.51 problem 1499

Internal problem ID [11502]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1499
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 }-\left (x^{2}-2 x \right ) y^{\prime \prime }-\left (x^{2}+\nu ^{2}-\frac {1}{4}\right ) y^{\prime }+\left (x^{2}-2 x +\nu ^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.032 (sec). Leaf size: 25

dsolve(x^2*diff(diff(diff(y(x),x),x),x)-(x^2-2*x)*diff(diff(y(x),x),x)-(x^2+nu^2-1/4)*diff(y(x),x)+(x^2-2*x+nu^2-1/4)*y(x)=0,y(x), singsol=all)
 

\[ y = {\mathrm e}^{x} c_{1} +c_{2} \sqrt {x}\, \operatorname {BesselI}\left (\nu , x\right )+c_3 \sqrt {x}\, \operatorname {BesselK}\left (\nu , x\right ) \]

✓ Solution by Mathematica

Time used: 0.126 (sec). Leaf size: 118

DSolve[(-1/4 + nu^2 - 2*x + x^2)*y[x] - (-1/4 + nu^2 + x^2)*D[y[x],x] - (-2*x + x^2)*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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