60.4.32 problem 1480

Internal problem ID [11483]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1480
Date solved : Tuesday, January 28, 2025 at 06:06:35 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime \prime }-\left (x +2 v \right ) y^{\prime \prime }-\left (x -2 v -1\right ) y^{\prime }+\left (x -1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.039 (sec). Leaf size: 54

dsolve(x*diff(diff(diff(y(x),x),x),x)-(x+2*v)*diff(diff(y(x),x),x)-(x-2*v-1)*diff(y(x),x)+(x-1)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.110 (sec). Leaf size: 114

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

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