60.4.35 problem 1483

Internal problem ID [11486]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1483
Date solved : Tuesday, January 28, 2025 at 06:06:36 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x y^{\prime \prime \prime }-4 \left (x +\nu -1\right ) y^{\prime \prime }+\left (2 x +6 \nu -5\right ) y^{\prime }+\left (1-2 \nu \right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 37

dsolve(2*x*diff(diff(diff(y(x),x),x),x)-4*(x+nu-1)*diff(diff(y(x),x),x)+(2*x+6*nu-5)*diff(y(x),x)+(1-2*nu)*y(x)=0,y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.075 (sec). Leaf size: 74

DSolve[(1 - 2*nu)*y[x] + (-5 + 6*nu + 2*x)*D[y[x],x] - 4*(-1 + nu + x)*D[y[x],{x,2}] + 2*x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

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