60.3.380 problem 1386

Internal problem ID [11390]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1386
Date solved : Monday, January 27, 2025 at 11:19:03 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=\frac {18 y}{\left (2 x +1\right )^{2} \left (x^{2}+x +1\right )} \end{align*}

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 58

dsolve(diff(diff(y(x),x),x) = 18/(2*x+1)^2/(x^2+x+1)*y(x),y(x), singsol=all)
 

\[ y = \frac {-36 c_{2} \left (x^{2}+x +1\right ) \arctan \left (\frac {\left (2 x +1\right ) \sqrt {3}}{3}\right )+16 c_{2} \left (x^{3}+x^{2}+\frac {11}{8} x +\frac {3}{16}\right ) \sqrt {3}+c_{1} \left (x^{2}+x +1\right )}{\left (2 x +1\right )^{2}} \]

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 80

DSolve[D[y[x],{x,2}] == (18*y[x])/((1 + 2*x)^2*(1 + x + x^2)),y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \exp \left (\int _1^x-\frac {3}{(2 K[1]+1) \left (K[1]^2+K[1]+1\right )}dK[1]\right ) \left (c_2 \int _1^x\exp \left (-2 \int _1^{K[2]}-\frac {3}{(2 K[1]+1) \left (K[1]^2+K[1]+1\right )}dK[1]\right )dK[2]+c_1\right ) \]