Internal problem ID [9102]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1523.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (x^{2}+1\right ) x^{3} y^{\prime \prime \prime }-\left (4 x^{2}+2\right ) x^{2} y^{\prime \prime }+\left (10 x^{2}+4\right ) x y^{\prime }-4 \left (3 x^{2}+1\right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.033 (sec). Leaf size: 26
dsolve((x^2+1)*x^3*diff(diff(diff(y(x),x),x),x)-(4*x^2+2)*x^2*diff(diff(y(x),x),x)+(10*x^2+4)*x*diff(y(x),x)-4*(3*x^2+1)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{2}+c_{2} x^{2} \left (\ln \relax (x )+1\right )+c_{3} \left (x^{3}+x \right ) \]
✓ Solution by Mathematica
Time used: 0.092 (sec). Leaf size: 40
DSolve[-4*(1 + 3*x^2)*y[x] + x*(4 + 10*x^2)*y'[x] - x^2*(2 + 4*x^2)*y''[x] + x^3*(1 + x^2)*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} x \left (c_2 (x-1)^2-2 c_1 ((x-3) x+1)+c_3 x+c_3 x \log (x)\right ) \\ \end{align*}