60.4.43 problem 1491

Internal problem ID [11494]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1491
Date solved : Monday, January 27, 2025 at 11:22:40 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }+3 x y^{\prime \prime }+\left (4 a^{2} x^{2 a}+1-4 \nu ^{2} a^{2}\right ) y^{\prime }&=4 a^{3} x^{2 a -1} y \end{align*}

✓ Solution by Maple

Time used: 0.106 (sec). Leaf size: 88

dsolve(x^2*diff(y(x),x$3)+3*x*diff(y(x),x$2)+(4*a^2*x^(2*a)+1-4*nu^2*a^2)*diff(y(x),x)=4*(a^3*x^(2*a-1))*y(x),y(x), singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.031 (sec). Leaf size: 102

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

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