60.4.8 problem 1456

Internal problem ID [11459]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1456
Date solved : Tuesday, January 28, 2025 at 06:06:32 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }+x^{2 c -2} y^{\prime }+\left (c -1\right ) x^{2 c -3} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.200 (sec). Leaf size: 73

dsolve(diff(diff(diff(y(x),x),x),x)+x^(2*c-2)*diff(y(x),x)+(c-1)*x^(2*c-3)*y(x)=0,y(x), singsol=all)
 

\[ y = x \left (c_{1} \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_{2} \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )^{2}+c_3 \operatorname {BesselJ}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right ) \operatorname {BesselY}\left (\frac {1}{2 c}, \frac {x^{c}}{2 c}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.017 (sec). Leaf size: 183

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

\[ y(x)\to c_1 \, _1F_2\left (\frac {1}{2}-\frac {1}{2 c};1-\frac {1}{c},1-\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right )+4^{-1/c} c^{-2/c} c_3 \left (x^{2 c}\right )^{\frac {1}{c}} \, _1F_2\left (\frac {1}{2}+\frac {1}{2 c};1+\frac {1}{2 c},1+\frac {1}{c};-\frac {x^{2 c}}{4 c^2}\right )+2^{-1/c} c^{-1/c} c_2 \left (x^{2 c}\right )^{\left .\frac {1}{2}\right /c} \, _1F_2\left (\frac {1}{2};1-\frac {1}{2 c},1+\frac {1}{2 c};-\frac {x^{2 c}}{4 c^2}\right ) \]