60.4.53 problem 1501

Internal problem ID [11504]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1501
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 }-2 \left (x^{2}-x \right ) y^{\prime \prime }+\left (x^{2}-2 x +\frac {1}{4}-\nu ^{2}\right ) y^{\prime }+\left (\nu ^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.041 (sec). Leaf size: 37

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

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

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 114

DSolve[(-1/4 + nu^2)*y[x] + (1/4 - nu^2 - 2*x + 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]
 

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