Internal problem ID [9891]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 4, linear fourth order
Problem number: 1568.
ODE order: 4.
ODE degree: 1.
CAS Maple gives this as type [[_high_order, _with_linear_symmetries]]
\[ \boxed {x^{4} y^{\prime \prime \prime \prime }+8 x^{3} y^{\prime \prime \prime }+12 x^{2} y^{\prime \prime }+a y=0} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 85
dsolve(x^4*diff(diff(diff(diff(y(x),x),x),x),x)+8*x^3*diff(diff(diff(y(x),x),x),x)+12*x^2*diff(diff(y(x),x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {c_{1} x^{-\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{2} x^{\frac {\sqrt {5-4 \sqrt {-a +1}}}{2}}+c_{3} x^{-\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}+c_{4} x^{\frac {\sqrt {5+4 \sqrt {-a +1}}}{2}}}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.009 (sec). Leaf size: 116
DSolve[a*y[x] + 12*x^2*y''[x] + 8*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {c_1 x^{-\frac {1}{2} \sqrt {5-4 \sqrt {1-a}}}+c_2 x^{\frac {1}{2} \sqrt {5-4 \sqrt {1-a}}}+c_3 x^{-\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}+c_4 x^{\frac {1}{2} \sqrt {4 \sqrt {1-a}+5}}}{\sqrt {x}} \]