60.4.58 problem 1508

Internal problem ID [11509]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1508
Date solved : Tuesday, January 28, 2025 at 06:06:42 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.022 (sec). Leaf size: 78

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

\[ y = x \left (c_{1} \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {\nu }{3}, -\frac {\nu }{3}+1\right ], -\frac {a \,x^{3}}{27}\right )+c_{2} x^{-\nu } \operatorname {hypergeom}\left (\left [\right ], \left [1-\frac {2 \nu }{3}, -\frac {\nu }{3}+1\right ], -\frac {a \,x^{3}}{27}\right )+c_3 \,x^{\nu } \operatorname {hypergeom}\left (\left [\right ], \left [1+\frac {\nu }{3}, \frac {2 \nu }{3}+1\right ], -\frac {a \,x^{3}}{27}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.363 (sec). Leaf size: 143

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

\[ y(x)\to 3^{-\nu -1} x a^{-\nu /3} \left (a^{\frac {\nu +1}{3}} \left (c_3 a^{\nu /3} x^{\nu } \, _0F_2\left (;\frac {\nu }{3}+1,\frac {2 \nu }{3}+1;-\frac {a x^3}{27}\right )+c_1 3^{\nu } \, _0F_2\left (;1-\frac {\nu }{3},\frac {\nu }{3}+1;-\frac {a x^3}{27}\right )\right )+\sqrt [3]{a} c_2 9^{\nu } x^{-\nu } \, _0F_2\left (;1-\frac {2 \nu }{3},1-\frac {\nu }{3};-\frac {a x^3}{27}\right )\right ) \]